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ARITHMETIC, 


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IN  MEMORIAM 
FLOR1AN  CAJ0R1 


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Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/adamsarithmOOadamrich 


REV 


ISED   AND  IMPROVEDlijDITION. 


ARITHMETIC: 

IN   TWO   PARTS. 
PART  FIRST, 

ADVANCED  LESSONS   IN  MENTAL  ARITHMETIC. 

PART   SECOND, 

RULES   AND  EXAMPLES  FOR  PRACTICE   IN 
WRITTEN    ARITHMETIC. 

FOR  CO'MMON  AND  RTGII  SCHOOLS. 


BY  FREDERIC  A.  ADAMS,  A.M., 

FORMER    PRINCIPAL    OF     DUMMER    ACADEMY. 


THIRTEENTH  THOUSAND. 


PHILADELPHIA: 
THOMAS,  COWPERTHWAIT  &  CO. 

New  York,  Roe  Lockwood  &.  Son  :  —Boston,  Phillips,  Sampson  <fc  Co ;  B.  B.  Mussev 

&  Co. ;  W.  J.  Reynolds  <k  Co. :— Baltimore,  Cushing  &  Bailey :— Charleston,  S.  C, 

McCarter  <&  Allen :  —  Louisville,  Ky.,  Morton  &  Griswold;  Beckwith  & 

Morton:  — St.  Louis,  Fisher  <fc  Bennett;  Wm.  D.  Skillman;  Amos  H. 

Shultz:  — Cincinnati,  J.  F.  Desilver:  — Nashville,  Wm.  T.  Berry; 

Chas.W.  Smith  :-Memphis,C.  C.Cleaves  :-Lexington,C.  S.  Bodley : 

—  Macon,  Geo ,  J.  M.  Boardman:—  Buffalo,  T.  &  M.  Butler. 

1851. 


ifrs-7 

Entered  accoflding  to  Act  of  Congress,  in  the  year  1848, 

BY  DANIEL  BIXBY, 

In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


RECOMMENDATIONS. 


From  Mr.  George  B.  Emerson,  Boston. 
I  have  carefully  examined  the  plan  of  Mr.  Adams's  work  on  Mental  Arithmetic, 
and  have  given  some  attention  to  its  execution  ;  and  I  am  confident  that  it  will  prove 
a  very  valuable  addition  to  the  means  of  instruction  in  Arithmetic.    It  is  a  suc-.| 
cessful  extension  of  the  admirable  method  of  Colburn's  First  Lessons,  with  such  i 
modifications  as  seemed  to  be  required  in  a  higher  work  on  the  same  general  model. 
It  occupies  unappropriated  ground  ;  and  it  deserves,  and  I  think  it  will  take,  a  high 
place  amongst  the  text-books.  GEO.  B.  EMERSON. 

From  Mr.  Thomas  Shcrwin,  Boston. 
I  have  carefully  examined,  in  manuscript,  the  work  of  Mr.  Adams  on  Mental  Arith- 
metic, and  am  much  pleased  with  it.  His  plan  is  good,  and  well  executed.  I  would, 
therefore,  heartily  recommend  his  book  to  Teachers  and  School  Committees,  as  one 
which  will  contribute  very  materially  to  the  attainment  of  that  very  important,  but 
much-neglected  branch  of  study,  Intellectual  Arithmetic. 

T.  SHERWIN,  Principal  of  Boston  English  High  School. 

From  Professor  Chase,  of  Dartmouth  College. 
Mr.  F.  A.  Adams.  *•».*,  ■       -  ; Hanover,  Oct.  12,  1846. 

My  Dear  Sir:  —  I  have  ■eJcaJniaed,  with  some*  c,a*e.;  your  treatise  on  Arithmetic, 
and  ajn  much  pleased  with  k.  '.The  practfce  aiid  halut  of  extending  mental  operations 
to  large  numbers  is  of  great  utility.  I  have  occasion,  very  frequently,  to  see  the 
inconvenience  that  yg^ng  njen 'suffer  /ran*,  -ike; want  of«  such  a  habit.  Not  less  valu- 
able than  the  habit  ©f.  operating  'mentalty  -upon  la/ge^humbers,  is  the  habit  of  per- 
forming the  more  advanced  operations"  oi*  arithmetic  without  the  aid  of  the  pencil. 

I  like  very  much,  also,  the  manner  in  which  you  have  treated  several  of  the  princi- 
ples which  you  have  developed ;  as,  for  example,  the  subject  of  the  common  divisor, 
the  least  common  multiple,  the  roots,  ratio,  and  proportion.  These  are  but  few  of  the 
subjects,  but  I  mention  them  as  examples. 

I  think  the  book  will  do  much  to  promote  the  proper  method  of  teaching  arithme- 
tic, —  by  demonstration  and  explanation.    I  am,  Dear  Sir,  very  truly  yours,  &c. 

S.  CHASE. 

From  Mr.  John  Tatlock,  Professor  of  Mathematics,  and  Mr.  A.  Hopkins,  Professor  of 
JYutural  Philosophy. 

Williams  College,  Nov.  GO,  1846. 
I  have  examined  a  treatise  on  Arithmetic  by  F.  A.  Adams,  and  am  much  pleased 
with  it.  I  think  it  well  adapted  to  teach  the  science  and  art  of  numbers,  and  at  the 
same  time  to  teach  the  art  of  thinking.  I  am  persuaded  that  a  thorough  training  in 
this  Arithmetic  would  prepare  students  for  the  farther  study  of  mathematics  betier 
than  nine  tenths  are  now  prepared. 
I  should  be  glad  if  every  student  who  enters  college  was  master  of  this  Arithmetic. 

JOHN  TATLOCK. 
A.  HOPKINS. 

In  School  Committee,  Roxburv,  Feb.  17,  1847. 
The  undersigned,  members  of  the  Committee  on  Text-Books,  recommend  that  the 
Arithmetic  prepared  by  Frederic  A.  Adams,  be  used  in  the  grammar  schools,  by  a!l 
the  classes  which  now  use  Leonard's  Arithmetic ;  and  that  hereafter  no  other  Arkh- 
me»ic  be  used  in  the  public  schools,  than  Adams's  together  with  Colburn's  First 
Lessons.  SAM'L.  H.  WALLEY,  Jr.        DAN'L.  LEACH, 

B.  E.  CUTTING.  THEO.  PARKER. 

GEO.  PUTNAM, 
In  School  Committee,  Feb.  17,  1847.  — The  above  recommendation  of  the  Cem- 
mittee  on  Text-Books,  was  this  day  adopted.  JOSHUA  SEAVER,  Secr'y. 

STEREOTYPED  AT  THE  BOSTON  TYPE  AND  8TEREOTYPE  FOUNDRY. 

Cajori 


PREFACE 


The  study  of  Arithmetic  in  the  schools  of  this  'country  received 
its  best  impulse,  unquestionably,  in  the  publication  of  Colburn's 
First  Lessons.  The  use  of  this  book,  and  of  others  made  on  a  sim- 
ilar plan,  has  done  much  towards  placing  this  branch  of  study  on  its 
proper  ground.  In  all  our  best  schools,  in  the  elementary  stages  of 
this  study,  the  logical  mode  has  taken  the  place  of  the  merely  for- 
mal; reason  is  the  guide,  instead  of  rule. 

The  want  of  a  higher  work  on  Mental  Arithmetic  has  long  been 
felt  by  teachers ;  and  by  business  men,  who  have  been  compelled, 
after  having  received  all  the  training  of  the  school-room,  to  adopt 
practical  modes  of  calculation  of  their  own,  to  meet  the  exigencies 
of  their  daily  business.  It  is  to  meet  these  wants  that  the  following 
work  has  been  prepared.     The  design  of  it  is,  — 

To  accustom  the  pupil  to  perform,  with  ease  and  readiness,  mental 
calculations  upon  somewhat  large  numbers  ;  — 

To  present  these  operations  in  their  natural  form,  freed  from  the 
inverted  and  mechanical  methods  which  belong  of  necessity  to 
operations  in  Written  Arithmetic ;  — 

To  train  the  student  to  such  a  power  of  apprehending  the  rela- 
tions of  numbers,  as  shall  give  him  an  insight  into  the  grounds  of 
the. rules  of  Arithmetic;  and,  consequently,  shall  release  him  from 
dependence  on  those  rules; — * 

To  prepare  the  members  of  our  schools,  when  they  shall  have  left 
school,  and  engaged  in  the  active  pursuits  of  life,  to  solve  mentally, 
and  with  ease  and  delight,  a  large  share  of  those  questions,  of  busi- 


4  PREFACE. 

ness  or  curiosity,  for  which  a  process  of  ciphering  is  usually  thought 
indispensable. 

The  "Advanced  Lessons"  presuppose,  of  course,  the  knowledge 
of  some  more  elementary  book.  To  study  this  work  successfully, 
therefore,  the  pupil  must  be  acquainted  with  Colburn's  First  Les- 
sons, or  some  other  work  occupying  essentially  the  same  ground. 

In  all  the  mental  calculations  in  large  sums,  it  will  be  found  a 
uniform  characteristic  of  this  work,  to  begin  with  the  highest  order 
of  numbers  in  the  sum,  —  hundreds  before  tens,  tens  before  units. 
In  this  way,  the  numbers  are  presented  in  the  same  order  in  which 
they  are  presented  in  the  common  usage  of  our  language.  In  most 
of  the  operations  of  Written  Arithmetic,  however,  the  smallest 
number  is  taken  first;  and  thus  a  method  is  pursued,  the  reverse 
ot  what  the  genius  of  our  language  would  naturally  suggest. 
Another  advantage  of  taking  the  highest  numbers  first,  in  Mental 
Arithmetic,  is,  that  we  thus  obtain  a  large  approximation  to  the 
final  answer,  at  the  first  step.  When  the  first  step,  however,  as 
in  written  addition  or  multiplication,  furnishes  only  the  units  of 
the  answer,  leaving  the  hundreds  or  thousands  still  unknown,  only 
a  minute  fraction  of  the  answer  is  at  first  obtained.  It  is  too  plain 
to  require  proof,  that  that  method  will  be  most  interesting  and 
gratifying  to  the  mind,  which  secures  the  largest  portion  of  the 
answer  at  the  first  step.  Another  advantage  of  the  method  here 
used,  is  found  in  the  fact,  that  we  naturally  make  the  higher 
order  the  standard;  and  the  lower  order  takes  its  value  in  the 
mind  from  a  comparison  with  the  higher,  as  a  certain  part  of  it. 
Thus  150  is  apprehended  by  the  mind,  as  one  hundred  and  half 
a  hundred.  This  is  not,  indeed,  the  method  of  acquiring  the 
idea  of  large  numbers,  but  the  method  of  combining  them  after 
the  idea  has  been  acquired ;  consequently,  it  is  the  legitimate 
method  of  instruction,  just  as  soon  as  the  pupil  is  qualified  to 
enter  on  the  study  of  such  combinations.  If,  now,  we  obtain  the 
number  of  the  highest  order  first,  we  have  a  standard,  under 
which  all  the  succeeding  orders  naturally  fall,  and  from  a  com- 
parison with  which  they  successively  take  their  value.  If  we 
begin  with  units,  however,  and  work  upward  through  the  higher 


PREFACE.  S> 

orders,  we  obtain  no  standard ;  we  must  hold  the  successive  num- 
bers in  suspense,  until  the  last  term  shall  furnish  the  nucleus  for 
the  group,  —  the  standard  under  which  all  the  lower  orders  shall 
take  their  rank. 

It  is  on  the  basis  of  these  facts,  which  are  only  indications  of 
the  laws  of  the  mind,  that,  throughout  the  mental  part  of  this 
Arithmetic,  the  author  has,  in  all  operations,  taken  the  highest 
order  of  numbers  first.  The  increased  interest  which  the  per- 
severing use  of  this  method  will  awaken  in  the  minds  of  pupils, 
will  be,  to  teachers,  a  better  commendation  of  its  correctness, 
than  any  more  extended  mental  analysis. 

Other  characteristic  features  of  the  Advanced  Lessons  are,  the 
extended  Multiplication  Table ;  the  use  of  the  complement  in 
addition;  the  analytical  treatment  of  fractions,  vulgar  and  decimal ; 
the  careful  separation  of  the  three  topics,  linear,  square,  and  solid 
measure  ;  the  construction  of  the  square  and  the  cube ; .  and  the 
mode  of  treating  proportion. 

The  Second  Part  contains  examples  in  Written  Arithmetic  on 
all  the  most  important  rules.  They  are  designed  to  be  sufficiently 
numerous  to  lead  the  student  to  ready  and  accurate  practice  in 
ciphering.  In  this  Part  the  author  has  aimed  to  interest  the 
scholar  by  furnishing  him  with  natural  and  reasonable  questions, 
and  to  aid  both  teacher  and  scholar  by  arranging  them  progres- 
sively. 

The  rules  and  explanations  will,  probably,  be  found  sufficient, 
after  a  thorough  mastery  of  the  First  Part.  It  is  not  necessary 
that  the  pupil  complete  the  First  Part  before  beginning  the 
Second.  He  may  carry  on  both  Parts  at  the  same  time;  but, 
under  each  particular  head,  the  mental  part  should  be  thoroughly 
mastered  before  the  written  examples  are  begun. 

The  answers  to  the  questions  in  the  Second  Part  are  given  In 
a  separate  work.  This  course  his  seemed  to  the  author,  on  the 
whole,  the  best,  notwithstanding  some  incidental  disadvantages 
that  may  arise  from  it.  It  will  enable  the  teacher  to  oversee  a 
much  larger  amount  of  work  in  Arithmetic  than  he  could  other- 
wise attend  to. 

1* 


6  PREFACE. 

The  Key  will  be  bound  up  with  the  Arithmetic,  for  the  use 
of  teachers ;  and  such  copies  will  be  lettered  "  Teacher's  Copy." 

The  present  contains  a  considerable  number  of  examples  more 
than  the  Third  Edition,  but  no  change  in  the  numbering  of  the 
sections  or  of  the  examples,  to  occasion  inconvenience  to  the 
teacher. 

To  aid  in  awakening  a  higher  interest  and  zeal  in  this  branch 
of  study,  the  author  will  offer  a  few  suggestions. 

Let  the  Key  be  used  as  little  as  the  teacher's  necessities  will 
permit. 

Let  original  questions  be  proposed  by  the  teacher  in  connection 
with  every  Section. 

Each  member  of  the  class  should  be  encouraged  to  propose 
original  questions  to  be  solved  by  the  class. 

It  will  often  be  "useful,  especially  in  a  review,  to  alter  some 
one  figure  in  the  conditions  of  each  question.  This  often  pro- 
duces a  happy  excitement,  and  gives  quite  a  new  zest  to  the 
study. 


REVISED    EDITION 


A  recent  revision  of  the  work  has  led  to  a  few  alter- 
ations in  the  First  Part,  designed  to  render  the  pupil's 
course  more  strictly  progressive.  A  thorough  acquaint- 
ance with  the  author's  First  Book  in  Arithmetic,  em- 
bracing the  "  lessons  for  practice  in  rapid  calculation," 
will  prepare  the  pupil  to  enter  successfully  on  the  study 
of  this  book.  It  is  indispensable,  however,  to  the  rapid 
progress  of  the  pupils,  that  they  commence  at  the  begin- 
ning, making  the  thorough  mastery  of  each  successive 
lesson  the  basis  of  the  studies  that  follow. 

Orange,  New  Jersey, 
September  1,  1850. 


CONTENTS 


PART    FIRST. 

Section.  Page. 

Explanations, 11 

I.    Multiplication  of  Tens  and  Units, 13 

II.    Multiplication  of  Tens  and  Units.     Complement, 16 

III.  Practical  Questions, 19 

IV.  Division, 21 

V.    Time.     Linear  Measure,  28 

VI.  Federal  Money.  Sterling  Money.  Dry  Measure. 
Avoirdupois  Weight.  Troy  Weight.  Apothecaries' 
Weight.      Cloth    Measure.       Wine    Measure.      Beer 

Measure.     Measure   of  the   Circle, 38 

VII.    Prime  Numbers, 50 

VIII.    Multiplication  and  Division  of  Fractions.     To  find  the 

Divisors  of  Numbers, 59 

IX.  Multiplication  of  Fractions  by  Fractions.  Division  of 
Fractions  by  Fractions.  To  Multiply  or  Divide  Whole 
Numbers  by  Fractions.    Addition  of  Fractions.    To  find 

a  Common  Denominator, G5 

X.    The  Least  Common  Multiple, 73 

XI.    Practical  Questions, 77 

XII.  Decimal  Fractions.  Addition  and  Subtraction  of  Deci- 
mals. Multiplication  of  Decimals.  Division  of  Deci- 
mals,     80 

XIII.  Reduction  of  Vulgar  Fractions  to  Decimals,   80 

XIV.  Interest.     Banking.     Discount.     Loss   and    Gain.     Per 

Centage, 93 

XV.    Square  Measure, ■ 101 . 

XVI.    Construction  of  the  Square.     Practical  Questions, 106 

XVII.    Practical  Questions  in  Square  Measure, 114 

XVIII.     Analysis  of  Problems, 119 

XIX.    Solid  Measure.     Construction  of  the  Cube, 122 

XX.    Ratio.     Proportion.     Comparison  of  Similar  Surfaces. 

Comparison  of  Similar  Solids, 120 

Notes  to  Part  First, 143 


CONTENTS. 


PART    SECOND 


I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

XI. 

XII. 

XIII. 

XIV. 

XV. 

XVI. 

XVII. 

XVIII. 

XIX. 

XX. 

XXI. 

XXII. 

XXIII. 

XXIV. 

XXV. 

XXVI. 

XXVII. 


XXVIII. 

XXIX.- 

XXX. 

XXXI. 

XXXII. 

XXXIII. 


Numeration  of  Whole  Numbers.     Numeration  of  De- 
cimals, ...» 147 

Addition, 150 

Subtraction, 152 

Multiplication, 155 

Division, 158 

Reduction, 163 

Reduction, 165 

Compound  Addition, 167 

Compound  Subtraction, '. 170 

Compound  Multiplication, 172 

Compound  Division, 173 

Miscellaneous  Examples, 174 

Divisibility  of  Numbers, 175 

Reduction  of  Fractions, 176 

Change  of  Numbers  and  Fractions  to  Higher  Terms,  179 

Multiplication  and  Division  of  Fractions, 180 

Multiplication  and  Division  of  Fractions, 182 

Addition  and  Subtraction  of  Fractions, 184 

Reduction  of  Denominate  Fractions, 186 

Change  of  Denominate  Integers  to  Fractions, 188 

Practical  Examples, 189 

Decimal  Fractions.     Addition  and  Subtraction.     Mul- 
tiplication and  Division, 190 

Reduction  of  Vulgar  Fractions  to  Decimals.     Repeat- 
ing and  Circulating  Decimals, 192 

Reduction  of  Denominate  Integers  to  Decimals, 194 

To  find  the  Integral  Value  of  Denominate  Decimals,  195 

Practical  Examples, 196 

Practical  Questions  in  Vulgar  and  Decimal  Fractions,  198 
Reduction  of  Currencies.     English  Currency.     Fed- 
eral Money  to  Sterling.     Canada  Currency.     New- 
England  Currency.     New  York  Currency.     Penn- 
sylvania Currency, 200 

Interest, , 203 

Partial  Payments.     Annual  Interest, 206 

Discount, 210 

Banking, 211 

Loss  and  Gain.     Per  Centage, 212 

Alligation, 216 


10 


CONTENTS. 


XXXIV. 

XXXV. 

XXXVI. 

XXXVII. 

XXXVIII. 

XXXIX. 

XL. 

XLI. 

XLII. 

XLIII. 

XLIV. 


XLV. 


XLVI. 


Equation  of  Payments,  • 220 

Square  Measure, •  •  •  •   222 

Duodecimals, 224 

Extraction  of  the  Square  Root, 226 

Extraction  of  the  Cube  Root, 230 

Proportion.     Practical  Questions.     Partnership,  . . .  233 
Arithmetical    Progression.        Descent    of    Falling 

Bodies, 243 

Geometrical  Progression, 248 

Mensuration  of  Surfaces, 250 

Mensuration  of  Solids, 252 

Miscellaneous  Theorems  and  Questions.  Specific 
Gravity.  Mechanical  Powers.  The  Lever.  The 
Wheel  and  Axle.  The  Screw.  Strength  of 
Beams  to  resist  Fracture.     Stiffness  of  Beams  to 

resist  Flexure, 254 

Business  Forms  and  Instruments.  Promissory 
Notes ;  —  on  Demand,  with  Interest ;  on  Time, 
with  Interest ;  on  Time,  without  Interest;  payable 
by  Instalments,  with  Periodical  Interest.  Remarks 
on  Promissory  Notes.  Receipts.  A  General 
Form  ;  for  Money  paid  by  another  Person ;  for 
Money  received  for  Another  ;  in  Part  of  a  Bond  ; 
for  Interest  due  on  a  Bond ;  on  Account ;  of  Pa- 
pers. Order  at  Sight.  Order  on  Time.  Award 
by  Referees.     Letter  of  Credit  for  Goods.    Power 

of  Attorney, 266 

On  the  Standard  of  Weights  and  Measures.  The 
English  System ;  adopted  by  the  Government  of 
the  United  States.  French  Decimal  System. 
French  Long  Measure.     French  Square  Measure. 

French  Decimal  Weight, 271 

Miscellaneous  Examples, 276 


EXPLANATIONS 


1    The  sign  =  indicates  equality;  as,  7  times  3  =  21. 

2.  The  sign  -|-  indicates  addition;  as,  15 -f- 7  =  22. 

3.  The  sign  — ,  placed  between  two  numbers,  indicates 
that  the  latter  number  is  to  be  taken  from  the  former;  as, 
9—4  =  5. 

The  larger  number  is  called  the  minuend ;  the  smaller,  the 
subtrahend. 

4.  The  sign  X  indicates  multiplication;  as,  6X7  =  42. 

The  two  numbers  are  called  factors ;  the  number  multi- 
plied is  called  the  multiplicand;  the  number  by  which  it  is 
multiplied,  the  multiplier. 

5.  The  sign  -£-  indicates  that  the  number  placed  before  it 
is  to  be  divided  by  the  number  after  it;  as,  15-^-5  =  3. 

The  number  to  be  divided  is  called  the  dividend;  the 
number  by  which  it  is  divided  is  called  the  divisor. 

6.  When  a  number  is  multiplied  by  itself,  the  product  is 
called  the  second  power  of  that  number,  or  the  square  of  it ; 
as,  2X2  =  4,  which  is  the  second  power,  or  the  square  of  2; 
so  9  is  the  square  of  3;  25,  the  square  of  5. 

7.  When  a  number  is  multiplied  by  itself,  so  as  to  be 
taken  3  times  as  a  factor,  the  product  is  called  the  third 
power  or  the  cube  of  the  number ;  thus,  8  is  the  cube  of  2, 
for  it  is  formed  by  multiplying  2X2X2;  27,  or  3X3X3,  is 
the  cube  or  third  power  of  3;  125,  or  5X5X5,  is  the  third 
power  of>6.  The  number  thus  used  as  a  factor,  is  called  the 
root  of  the  power;  thus,  3  is  the  square  root  of  9,  and  the 
cube  root  of  27 :  5  is  the  square  root  of  25. 


12 


EXPLANATIONS. 


The  number  of  the  power  may  be  expressed  by  a  small 
figure;  thus,  23  is  the  3d  power  of  2;  32  is  the  2d  power  of 
3;  53  is  the  3d  power  of  5. 

An  angle  is  formed  when  two  lines  meet,  run- 
ning in  different  directions. 

A  triangle  is  a  figure  bounded  by  three  straight 
lines.  It  is  called  a  triangle,  because  it  has  three 
angles.  An  equilateral  triangle  has  all  its  sides 
equal. 

A  rigid  angle  is  formed  when  one  line  meets 
_  another,  making  the  angles  on  both  sides  equal. 


A  square  is  a  four-sided  figure,  the  sides  of  which 
are  all  equal,  and  the  angles  of  which  are  right  angles. 
The  diagonal  divides  it  into  two  equal  parts. 

A  rectangle  is  a  four-sided  figure,  the  oppo- 
site sides  of  which  are  equal,  and  the  angles  of 
which  are  right  angles.  The  diagonal  divides 
it  into  two  equal  parts. 

A  parallelogram  is  a  four-sided  figure,  the 
opposite  sides  of  which  are  equal  and  parallel. 
The  diagonal  divides  it  into  two  equal  parts. 

A  circle  is  a  figure  bounded  by  a  curved  line, 
called  the  circumference,  every  part  of  which  is 
equally  distant  from  the  centre. 

A  straight  line  from  the  centre  to  the  circumference  is 
called  the  radius. 

The  diameter  is  a  line  drawn  from  side  to  side  of  the  cir- 
cle, through  the  centre.  It  follows  that  the  diameter  is  equal 
to  twice  the  radius. 

Any  portion  of  the  circumference,  considered  by  itself,  is 
called  an  arc. 

A  sector  of  a  circle  is  a  portion  of  it  bounded  by  two  radii 
and  the  arc  between  them. 

A  sphere  is  a  solid  bounded  by  a  curved  surface  every  part 
of  which  is  equally  distant  from  the  centre  of  the  solid. 


O 


MENTAL   ARITHMETIC. 


PART    FIRST 


SECTION   I. 


MULTIPLICATION   OF  TENS  AND   UNITS. 

1.  A  man  drove  six  oxen  to  market,  and  sold  three 
of  them  for  50  dollars  apiece.  What  did  they  come 
to? 

Three  times  50  are  150.     Ans.  150  dollars. 

He  sold  the  remaining  three  for  52  dollars  apiece. 
What  did  they  come  to? 

Three  times  50  are  150,  and  three  times  2  are  6, 
which  added  to  150  makes  156.     Ans.  156  dollars. 

What  did  they  all  come  to  ? 

Twice  100  is  200,  and  twice  50  is  100,  which 
added  to  200  makes  300,  and  6  added  to  300  makes 
306.     Ans.  306  dollars. 

2.  A  merchant  bought  45  barrels  of  flour  for  6  dol- 
lars a  barrel.     What  did  it  come  to  ? 

6  times  40  are  240,  and  6  times  5  are  30 ;  30  added 
to  240  makes  270.     Ans.  270  dollars. 

He  bought  75  barrels  more  at  5  dollars  a  barrel. 
What  did  it  come  to? 

5  timers  70  are  350 ;  5  times  5  are  25,  which  added 
to  350  makes  375.     Ans.  375  dollars. 

What  did  all  the  flour  come  to  ? 
2 


14 


MENTAL    ARITHMETIC. 


;300  aud  200  are  5.00;  70  and  70  are  140,  which 
added  to  500  makes  640,  and  5  are  645.  Aus.  645 
dollars. 

3.  What  will  87  barrels  of  flour  come  to  at  6  dol- 
lars a  barrel  ? 

6  times  80  are  480;  and  6  times  7  are  42,  which 
added  to  480   makes  522.     Ans.  522  dollars. 

4.  What  are  7  times  68  ?     What  are  8  times  72  ? 
What  are  9  times  84  ?     What  are  4  times  96  ? 

8  times  64  ?  5  times  72  ?  7  times  83  ?  5  times  79  ? 
4  times  98  ?  3  times  81  ?  6  times  73?  6  times  86  ? 

The  preceding  examples  will  show  the  importance 
of  being  able  readily  to  multiply  tens  by  units.  This 
becomes  easy  after  acquiring  the  Multiplication  Table. 
It  may  be  connected  with  a  review  of  the  Multiplica- 
tion Table  in  the  following  manner : 


Twice  1  are  how  many? 
Twice  2  are  how  many? 
Twice  3?  Twice  30? 
Twice  5?  Twice  50? 
Twice  7  ?  Twice  70  ? 
Twice  9?       Twice  90? 


Twice  10  are  how  many? 
Twice  20  are  how  many? 
Twice    4?        Twice     40? 
Twice    6?        Twice     60? 
Twice    8?        Twice     80? 
Twice  10?        Twice    100? 


3  times  1  ? 
3  times  3  ? 
3  times  5? 
3  times  7? 

3  times  9  ? 

4  times  1  ? 
4  times  3  ? 
4  times  5  ? 
4  times  7  ? 

4  times  9  ? 

5  times  1  ? 
5  times  3? 
5  times  5  ? 
5  times  7  ? 
5  times  9  ? 


3  times  10? 
3  times  30  ? 
3  times  50  ? 
3  times  70 ! 

3  times  90  ? 

4  times  10? 
4  times  30  1 
4  times  50  ? 
4  times  70  ? 

4  times  90  ? 

5  times  10  ? 
5  times  30 ! 
5  times  50? 
5  times  70  ? 
5  times  90  ? 


3  times  2  ? 
3  times  4  ? 
3  times    6  ? 

3  times  8  ? 
:>,  times  10  ? 

4  times  2  ? 
4  times  4? 
4  times  6? 
4  times  8  ? 

4  times  10? 

5  times  2  ? 
5  times  4? 
5  times  6? 
5  times  8  ? 
5  times  10? 


3  times  20  ? 
3  times  40  ? 
3  times  60  ! 
3  times    80  ( 

3  times  100? 

4  times  20 1 
4  times  40  ? 
4  times  60  ! 
4  times  80  ? 

4  times  100  ? 

5  times  20  ? 
5  times  40  'i 
5  times  60  ? 
5  times  80  ? 
5  times  100? 


MULTIPLICATION    OF    TENS    AND    UNITS. 


15 


0  times 

1? 

6  times 

10? 

6  times    2  ? 

6  times    20  ? 

6  times 

3? 

6  times 

30? 

6  times    4  ? 

•6  times    40? 

6  times 

5? 

6  times 

50? 

6  times    6  ? 

6  times    60  ? 

6  times 

7? 

6  times 

70? 

6  times    8  ? 

6  times    80  ? 

6  times 

9? 

6  times 

90? 

6  times  10? 

G  times  100  ? 

7  times 

1? 

7  times 

10? 

7  times    2? 

7  times    20  ? 

7  times 

3? 

7  times 

30? 

7  times    4  ? 

7  times    40  ? 

7  times 

5? 

7  times 

50? 

7  times    6? 

7  times    60  ? 

7  times 

7? 

7  times 

70? 

7  times    8  ? 

7  times    80? 

7  times 

9? 

7  times 

90? 

7  times  10? 

7  times  100? 

8  times 

1? 

8  times 

10? 

8  times    2  ? 

8  times    20  ? 

8  times 

3? 

8  times 

30? 

8  times    4  ? 

8  times    40  ? 

8  times 

5? 

8  times 

50? 

8  times    6  ? 

8  times    60  ? 

8  times 

7? 

8  times 

70? 

8  times    8  ? 

8  times    80  ? 

8  times 

9? 

8  times 

90? 

8  times  10  ? 

8  times  100? 

9  times 

1? 

9  times 

10? 

9  times    2  ? 

9  times    20  ? 

9  times 

3? 

9  times 

30? 

9  times    4  ? 

9  times    40  ? 

9  times 

5? 

9  times 

50? 

9  times    6? 

9  times    GO  I 

9  times 

7? 

9  times 

70? 

9  times    8  ? 

9  times    80  ? 

9  times 

9? 

9  times 

90? 

9  times  10  ? 

9  times  100? 

10  times 

1? 

10  times 

10? 

10  times    2  ? 

10  times    20? 

10  times 

3? 

10  times 

30? 

10  times    4  ? 

10  times    40? 

10  times 

5? 

10  times 

50? 

10  times    6? 

10  times    60? 

10  times 

7? 

10  times 

70? 

10  times    8  ? 

10  times    80  ? 

10  times 

9? 

10  times 

90? 

10  times  10? 

10  times  100  ? 

1 1  times 

1? 

11  times 

10? 

11  times    2? 

It  times    20? 

11  times 

3? 

11  times 

30? 

11  times    4? 

11  times    40? 

11  times 

5? 

11  times 

50? 

11  times    6? 

11  times    60? 

11  times 

7? 

11  times 

70? 

11  times    8? 

1 1  times    80  ? 

11  times 

9? 

11  times 

90? 

11  times  10? 

11  times  100? 

11  times  11? 

11  times 

110? 

11  times  12? 

11  times  120? 

12  times 

1? 

12  times 

10? 

12  times    2? 

12  times    20  ? 

12  times 

3? 

12  times 

30? 

12  times    4  ? 

12  times    40? 

12  times 

5? 

12  times 

50? 

12  times    6  ? 

12  times    60? 

12  times 

7? 

12  times 

70? 

12  times    8? 

12  times    80? 

12  times 

9? 

12  times 

90? 

12  times  10? 

12  times  100? 

12  times  ] 

LI? 

12  times  110? 

12  times  12? 

12  times  120  ? 

16  MENTAL    ARITHMETIC. 

A  number  which  contains  another  number  a  certain 
number  of  times,  is  a  multiple  of  that  number. 
Thus  6  is  a  multiple  of  2  ;   15  of  3 ;  28  of  7.* 

Name  all  the  multiples  of  2,  from  2  to  60. 

Name  the  multiples  of  20,  from  20  to  600. 

What  are  the  multiples  of  3  up  to  75  ?  Of  30  up  to 
750? 

What  are  the  multiples  of  4  up  to  80  ?  Of  40  up  to 
800? 

What  are  the  multiples  of  5  up  to  100?  Of  50  up 
to  1000?  Of  6  to  72?  Of  60  to  720?  Of  7  to  84?  Of 
70  to  840  ?  Of  8  to  96  ?  Of  80  to  960  ?  Of  9  to  108  ? 
Of  90  to  1080  ?    Of  10  to  120  ?   Of  100  to  1200  ? 


SECTION    II. 

MULTIPLICATION  OF  TENS  AND  UNITS.— COMPLEMENT. 

1.  What  will  17  tons  of  hay  come  to  at  8  dollars  a 
ton? 

Ans.  8  times  10  are  80,  and  8  times  7  are  56  ;  56 
added  to  80  makes  136.     136  dollars. 

2.  What  will  37  pounds  of  sugar  come  to  at  9  cents 
a  pound  ?     J 

3.  A  man  drove  87  sheep  to  market,  and  sold  them 
for  6  dollars  apiece.     What  did  they  come  to  ? 

4.  A  man  travelled  on  foot  eight  days ;  he  travelled 
29  miles  each  day.  How  many  miles  did  he  travel  in 
all?     i, 

In  each  of  the  above  examples,  the  second  product, 
when  added  to  the  first,  makes  a  sum  exceeding  the 
next  even  hundred  :  thus,  in  the  1st  ex.,  80  +  56  ;  in 

*  See  Note  1 ,  at  the  end  of  Part  First 


MULTIPLICATION    OF    TENS    AND    UNITS.  17 

the  2d,  270  +  63;  in  the  3d,  480  +  42;  in  the  4th, 
160  +  72. 

In  order  to  perform  snch  examples  with  ease,  quick- 
ness, and  without  mistake,  each  step  in  the  process 
should  be  made  the  subject  of  distinct  practice.  To 
illustrate  these  steps  by  the  first  example,  80  +  56,  the 
first  thing  to  be  done  is  to  think  of  the  number  which 
must  be  added  to  80  to  make  100,  namely,  20 ;  the 
next  is  to  take  this  20  from  the  56,  and  what  remains 
(36)  will  belong  to  the  next  hundred. 

The  number  which,  in  such  cases,  must  be  added  to 
a  given  number  to  make  up  an  even  hundred,  may 
be  called  the  complement  of  that  number.  Thus  the 
complement  of  80  is  20  ;  of  60,  40 ;  of  90,  10  ;  of  56, 


44. 

What 

is  the 

compl 

ement  of  10 

?  30? 

50?   ' 

ro? 

# 

What  is  the  complement 

of 

10? 

20? 

30? 

40? 

50? 

60? 

70? 

80? 

90? 

11  \ 

21? 

HI? 

41? 

51? 

61? 

71? 

81? 

91? 

12? 

22? 

32? 

42? 

52? 

62? 

72? 

82? 

92? 

13? 

23? 

33? 

43? 

53? 

63? 

73? 

83? 

93? 

14? 

24? 

34? 

44? 

54? 

64? 

74? 

84? 

94? 

15? 

25? 

35? 

45? 

55? 

65? 

75? 

85? 

95? 

16? 

26? 

36? 

46? 

561 

66? 

76? 

86? 

96? 

17? 

27? 

37? 

47? 

57? 

67? 

77? 

87? 

97? 

18? 

28? 

38? 

48? 

58? 

68? 

78? 

88? 

98? 

19? 

29? 

39? 

49? 

59? 

69? 

79? 

89? 

99? 

How  many  are  40  +  76?  80  +  34?  70  +  91?  90  +  17? 
25  +  83?  36  +  71?  45  +  82?  56  +  73?  43  +  82?  95  +  36? 
37  +  84?  45  +  76?  88  +  37?  94  +  17?  76  +  87? 

f  How  many  are  How  many  are  $ 

12X2,  3,  4,  5,  6,  7,  8,  9,  10?  16X2,  3,  4,  5,  6,  7,  8,  9,  10? 

13X2,  3,  4,  5,  6,  7,  8,  9, 10  ?  17X2,  3,  4,  5,  6,  7,  8,  9,  10  ? 

14X2,  3,  4,  5,  6,  7,  8,  9,  10  ?  18x2,  3,  4,  5,  6,  7,  8,  9,  10  ? 

15X2,  3,  4,  5,  6,  7,  8,  9,  10?  19X2,  3,  4,  5,  6,  7,  8,  9,  10? 


See  Note  2.  t  Note  3. 

2*  R 


18 


MENTAL    ARITHMETIC. 


How  many  are 

20X2,3,4,5,6,7,8 

4 


How  many  are 


21X2 
22X2 
23X2 
24X2 
25X2 
26X2 
27X2 
28X2 
29X2 
30X2 
31X2 
32X2 
33X2 
34X2 
35X2 
36X2 
37X2 
38X2 
39X2 
40X2 
41X2 
42X2 
43X2 
44X2 
45X2 
46X2 
47X2 
48X2 
49X2 


9,  10?)<  50X2 
9,10?       51X2 


9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,  10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,  K)? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 
9,10? 


52X2 
53X2 
54X2 
55X2 
56X2 
57X2 
58X2 
59X2 
60X2 
61X2 
62X2 
63X2 
64X2 
65X2 
66X2 
67X2 
68X2 
69X2 
70X2 
71X2 
72X2 
73X2 
74X2 
75X2 
76X2 
77X2 
78X2 
79X2 


,  3,  4,  5,  6 

,7 

»8 

A 

,3,4,5,6 

7 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9< 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6, 

7 

8 

9, 

,  3,  4,  5,  6, 

7' 

8 

9, 

,  3,  4,  5,  6, 

7, 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6, 

7 

8 

9, 

,  3,  4,  5,  6, 

~, 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6, 

7 

8 

9, 

■  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6, 

7, 

8, 

9„ 

,  3,  4,  5,  6, 

7 

8 

9, 

,  3,  4,  5,  6, 

7 

8 

9, 

,  3,  4,  5,  6 

7 

8 

9, 

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7, 

8, 

9, 

,  3,  4,  5,  6, 

7 

6 

9, 

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7 

8 

9, 

,  3,  4,  5,  6, 

7, 

e, 

9, 

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7 , 

s, 

9, 

,  3,4,  5, 6, 

7, 

8 

9, 

,  3,  4,  5,  6, 

7; 

8 

9, 

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7 

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9, 

,  3,  4,  5,  6 

7 

8 

9, 

,  3,  4,  5,  6, 

", 

8, 

9, 

To  multiply  any  number  less  than  10  by  11,  repeat 
the  figure  expressing  the  number  ;  as,  3  times  11  is  33, 
4X11  =  44. 

To  multiply  by  11  any  number  of  two  figures. 
Think  of  the  first  figure,  then  of  the  sum  of  the  two 


PRACTICAL    QUESTIONS.  19 

figures,  then  of  the  last  figure.  These  three  figures 
will  express  the  answer.  Thus,  11X23;  the  first,  2; 
the  sum  of  the  two,  5;  the  last,  3.  A?is.  253.  11X 
24  =  264,  11X32  =  352,  11X43  =  473. 

Remember,  if  the  sum  of  the  two  is  as  much  as  10, 
you  must  increase  the  first  figure  by  1. 

How  many  are  11X26?  11X28?  11X29?  11X41? 
11X43?  11X45?  11X61?  11X62?  11X64?  11X71? 
11X73?  11X81?  11X94?  11X75?  11X86?  11X89? 
11X82?    11X84? 


SECTION    III. 

PRACTICAL   QUESTIONS. 

1.  If  a  railroad  car  travels  23  miles  in  one  hour, 
how  far  will  it  travel  in  9  hours  ? 

2.  If  a  horse  travels  38  miles  in  one  day,  how  far 
will  he  travel  in  6  days  ? 

3.  If  a  man  earns  14  dollars  a  month,  how  much 
will   he  earn  in  7  months? 

4.  If  a  man  spends  6  cents  a  day  for  ardent  spirit, 
how  much  will  that  amount  to  in  10  days  ?  How 
much  in  30  days  ?  How  much  in  300  days  ?  How 
much  in  60  days?  How  much  in  5  days?  How 
much   in  365  days  ? 

5.  If  a  man  earns  10  cents  in  an  hour,  and  works  12 
hours  in  a  day,  how  much  will  he  earn  in  a  week,  there 
being  6  working  days  in  a  week  ?  How  much  in  10 
weeks  ?     How  much  in  50  weeks  ? 

6.  If  a  scholar  in  school  is  idle  18  minutes  in  the 
forenoon,  and  18  minutes  in  the  afternoon,  how  much 
time  will  he  lose  in  a  week,  if  there  are  6  forenoons 
and  4  afternoons  of  school  time  in  a  week  ? 

7.  If  a  town  is  6  miles  long,  and  5  miles  broad,  how 


26  MENTAL    ARITHMETIC. 

many  square  miles  does  it  contain?  If  there  are  40 
inhabitants  on  every  square  mile,  how  many  inhab- 
itants does  the  town  contain?    40  times  30.     4  times 

30  are  120.  40  times  30  are  10  times  as  many.  If 
one  in  12  of  the  inhabitants  were  able-bodied  men, 
how  many  able-bodied  men  would  there  be  ?  If  one 
in  6  are  able-bodied  men,  how  many  such  are  there? 

8.  What  will  146  yards  of  broadcloth  come  to  at 

5  dollars  a  yard? 

9.  What  will  86    yards  of  broadcloth  come  to  at 

6  dollars  and  a  half  a  }-ard  ?    . 

10.  What  will  710  barrels  of  rlour  come  to  at  6  dol- 
lars a  barrel  ?  At  5  dollars  a  barrel  ?  At  5  dollars  and 
a  half  a  barrel  ? 

11.  What  will  33  gallons  of  molasses  come  to  at 

31  cents  a  gallon  ?  At  34  cents  a  gallon  ?  At  40  cents 
a  gallon  ? 

12.  What  will  38  pounds  of  coffee  come  to  at 
14  cents  a  pound?  £At   16  cents  a  pound  ?  U  ^ 

13.  If  there  are  12  windows  in  the  front  of  a  fiouse, 
10  in  the  rear,  and  6  in  each  end,  each  window  contain- 
ing 12  lights ;  how  many  lights  are  there  in  the  front  of 
the  house?  How  many  in  the  rear?  How  many  in 
both  ends?  How  many  in  all?  If  each  light  cost  10 
cents,  how  much  would  they  all  cost  ?  If  the  setting  of 
each  light  cost  5  cents,  how  much  would  the  setting  of 
them  all  cost? 

14.  There  are  8  houses  in  a  row:  each  house  has  8 
windows  in  front,  7  in  the  rear,  and  5  in  each  end,  each 
window  containing  12  lights ;  how  many  lights  are 
there  in  each  house?  How  many  in  ajl  the  houses? 
What  would  be  the  cost  of  them  all  at  "If  cents  for  each 
light  ?  What  would  be  the  cost  of  setting  them  all  at  3 
cents  for  each  light?  At  the  above  price,  what  would  be 
the  cost  of  the  glass  and  the  setting  of  it  for  one  of  the 
houses?  What  would  be  the  cost  of  the  glass  and  the 
setting  of  it  for  two  of  the  houses  ?  For  three  of  the 
houses  ?    For  four  of  the  houses  ?   For  five  of  the  houses? 


DIVISION.  21 

15.  What  are  8£  tons  of  hay  worth,  at  13  dollars  a 
ton? 

16.  If  one  acre  of  ground  produce  65  bushels  of 
corn,  how  much  would  grow  on  9  acres  ? 

\1.  If  an  acre  of  ground  produce  228  bushels  of 
potatoes,  how  many  bushels  would  grow  on  5  acres  ? 

18.  If  standing  wood  is  worth  2  dollars  a  cord,  what 
is  the  value  of  the  wood  on  7  acres,  each  of  which 
furnishes  18  cords? 

19.  If  there  are  200  families  in  a  town,  and  each 
family  consumes  12  cords  of  wood  annually,  how 
many  cords  are  used  in  the  town  each  year? 

What  is  the  whole  value  of  the  wood,  at  3£  dollars 
a  cord  ?  How  much  money  will  be  saved  in  the  town 
if  each  family  burns  2  cords  less  than  before  ? 


SECTION    IV. 

DIVISION. 

1.  What  is  one  half  of  20?  Of  40?  Of  60?  Of 
80?    Of  100?    Of  120?    Of  140?    Of  160? 

2.  What  is  one  half  of  22  ?  Of  42  ?  Of  62  ?  Of  82 1 
Of  102  ?  Of  112  ?  Of  122  ?  Of  142  ?  Of  162  ?  Of  182  ? 

3.  What  is  one  half  of  44  ?  64  ?  S6  ?  48  ?  66  ?  28  ? 
84?  68?  46?  24?  26?  62? 

1      4.  What  is  one  half  of  70  ?  Divide  it  into  60  and  10. 

What  is  one  half  of  90?    Divide  it  into  80  and  10. 

What  is  one  half  of  50 ?  Of  30?  Of  110?  Of  130? 
Of  150? 

5.  What  is  one  half  of  32  ?  Of  54  ?  Divide  it  into 
50  and  4. 

What  is  one  half  of  76  ?  Of  74  ?  Of  78  ?  Of  96  ? 
Of9Sr  Of  92?  Of  94?  Of  72?  Of  76 ?  Of  58?  Of  56? 


22  MENTAL    ARITHMETIC. 

6.  What  is  one  half  of  43?  One  half  of  40  is  20. 
One  half  of  3  is  H ;  this  added  to  20  makes  21*. 

What  is  one  half  of  47  ?  Of  49  ?  Of  63  ?  Of  65  ? 
Of  67  ?    Of  69  ?    Of  83  ?    Of  85  ?    Of  87  ?    Of  89  ? 

7.  What  is  one  half  of  33  ?    Divide  into  30  and  3. 
What  is  one  half  of  35  ?    37  ?    39  ?    51  ?   53  ?  55  ? 

57}    59?    Of  71?    Of  73?    Of  75 ?    Of  77 ?    Of  79  ? 
Of  91?    Of  93?    Of  95?   Of  97?    Of  99 ? 

8.  What  is  one  half  of  367?  Divide  into  300, 
60,  and  7. 

What  is  one  half  of  674  ?  Of  895  ?  Of  724  ?  Of 
632?    Of  945?    Of  424?    Of  688?    01546?    Of  392? 

We  can  now  find  a  very  quick  way  of  multiplying 
any  number  by  5.  Take  one  half  the  number :  mut 
tiply  that  by  10.*  We  will  take  the  numbers  in  ques- 
tion 2,  and  multiply  them  by  5  in  this  way. 

Multiply  22  by  5.  Half  of  22  is  11,  and  ten  times 
11  is  110. 

Multiply  42  by  5.  Half  of  42  is  21  ;  ten  times  that 
is  210. 

Multiply  62  by  5.     Half  of  62  is  31 ;  310. 

Multiply  82  by  5.     Half  is  41  ;  410. 

Multiply  102  by  5.     Half  is  51 ;  510. 

Multiply  112  by  5.     Half  is  56;  560. 

Multiply  122  by  5.     Half  is  61 ;  610. 

Multiply  142  by  5.     Half  is  71  ;  710. 

Multiply  162  by  5.     Half  is  81 ;  810. 

Multiply  182  by  5.     Half  is  91 ;  910. 

9.  Multiply  by  5,  in  this  way,  the  numbers  in  ques- 
tion 3;  44;  64;  86;  48;  66;-28;  84;  68;  46;  24; 
26;  62. 

You  can,  if  you  wish,  perform  these  examples  by 
both  methods,  and  thus  prove  the  work  correct. 

Multiply  862  by  5.  Half  is  431;  4310.— Multiply 
672  by  5.     Half  is  336  ;  3360. 

10.  Multiply  6S6  by  5;  748  by  5;  932  by  5;  896 
by  5 ;   1262  by  5. 


division.  23 

If  the  number  to  be  multiplied  is  an  odd  number, 
so  that  half  of  it  will  show  the  fraction  £,  this,  when 
you  multiply  by  10,  will  become  5 ;  for  teu  halves 
are  5. 

Multiply  781  by  5.     Half  is  3904- ;  3905,  Ans. 

11.  Multiply  963  by  5.  Half  is  481J;  4815.— 
Multiply  845  by  5;  3S1  by  5;  953  by  5;  845  by  5; 
637  by  5;  429  by  5. 

12.  What  is  one  fourth  of  40?  One  fourth  of  80? 
One  fourth  of  120?  One  fourth  of  12  is  3 ;  a  fourth 
of  120  is  10  times  as  much  ;  30.  — What  is  one  fourth 
of  160?  One  fourth  of  200?  Of  240?  Of  280?  Of 
320?    Of  360?    Of  400. 

13.  What  is  one  fourth  of  60  ?  Take  half  of  it ; 
then  half  of  that  half.  Half  of  60  is  30 ;  half  of  30 
is  15. — What  is  one  fourth  of  100?  One  fourth  of 
140?  One  fourth  of  180?  One  fourth  of  220?  One 
fourth  of  260  ?    One  fourth  of  300  ? 

14.  Another  way  of  finding  one  fourth  of  the  num- 
bers in  the  last  example,  is  as  follows : 

What  is  one  fourth  of  60  ?  Divide  60  into  40  and 
20.    One  fourth  of  40  is  10  ;  one  fourth  of  20  is  5 ;   15. 

What  is  one  fourth  of  100?     Divide  into  80  +  20. 

What  is  one  fourth  of  140?     Divide  into  120  +  20. 

What  is  one  fourth  of  180?  Divide  into  160  +  20, 
&c. 

15.  What  is.  one  fourth  of  30?    Of  50?    Of  70? 
Find  the  best  way  of  answering  these,  for  yourself. 
What  is  one  fourth  of  90?    Of  110?    Of  130?     Of 

150?    Of  170?    Of  190?    Of  210?    Of  230?    Of  250. 

16.  What  is  one  fourth  of  76  ?  Divide  the  number 
into  40  and  36.  —  What  is  one*  fourth  of  96  ?  Divide 
the  number  into  80  and  16. 

What  is  one  fourth  of  52  ?    Of  64  ?    Of  84  ? 

17.  What  is  one  fourth  of  368  ?  There  are  several 
ways  of  dividing  this  number ;  first,  into  200  + 100 
+  6(\+  8 ;  a  second  way  would  be,  into  200  +  160  +  8 j 


24  MENTAL    ARITHMETIC. 

another  way,  into  320 -|- 48.  This  is  shorter  than 
either  of  the  former.  A  better  division  still  is  into 
360  +  8. 

What  is  one  fourth  of  496?  Into  what  different 
sets  of  numbers,  each  divisible  by  4,  can  you  divide 
this  ?  What  is  one  fourth  of  964  ?  Of  336  ?  Of  836  ? 
596?    472?    1324?     1728?    2236? 

18.  What  is  one  tenth  of  10?  Of  20?  Of  30? 
Of  40?  Of  50?  Of  60?  70?  80?  90?  100?  110? 
120?     130?     140?     150? 

19.  What  is  one  tenth  of  5?  Ans.  5  tenths  of  1,  or 
yV,  equal  to  £. 

What,  then,  is  one  tenth  of  15?  Of  25?  35?  45? 
55}  65?  75}  85?  95?  14?  17?  36?  47?  52? 
91?    43?    28?    65?    86?    47? 

20.  What  is  one  fifth  of  25?  40?  45?  50?  55} 
60?  65?  70?  Divide  70  into  50  and  20.  — Of  75} 
Divide  into  50  and  25.  — Of  80?  Divide  into  50 
and  30.  — Of  85?    Of  90?    Of  95?    Of  100? 

21.  What  is  one  fifth  of  64?  Of  82?  Of  91?  Of 
67?  Of  73?  Of  59?  Of  63?  Of  72?  Of  78?  Of 
83?    Of  87? 

22.  What  is  one  fifth  of  140?  Of  385?  Of  260? 
Of  480?  Of  390?  Of  580?  Of  470?  Of  865?  Of 
395? 

23.  The  following  is  a  short  way  of  dividing  a 
number  by  5.  Take  one  tenth  of  the  number,  and 
double  it.  That,  of  course,  gives  2  tenths,  which  is 
equal  to  1  fifth.  Take  the  numbers  in  the  last  ex- 
ample, and  divide  by  5  in  this  way.  One  fifth  of  140  ; 
one  tenth  is  14;  double  that  is  28.  One  fifth  of  385; 
one  tenth  is  38  and  5  tenths ;  twice  that  is  77.  One 
fifth  of  260 ;  one  tenth  is  26 ;  twice  26  is  52.  One 
fifth  of  480 ;  one  tenth  is  48 ;  twice  that  is  96.  What 
is  one  fifth  of  390?    Of  580?    470?    865?    395? 

24.  The  following  is  a  short  way  of  multiplying  a 
number  by  25.     Take   one  fourth   of   the   number; 


DIVISION.  25 

multiply  that  by  100.  This  will  give  100  fourths, 
which  are  equal  to  25  whole  ones. 

Multiply  40  by  25.  One  fourth  is  10 ;  one  hundred 
times  that  are  1000.  —  Multiply  60  by  25.  One  fourth 
is  15  ;   1500. 

Multiply    80  by  25.     One  fourth  is  20 ;  2000. 

Multiply  120  by  25.     A  fourth  is  30;  3000. 

Multiply  112  by  25.     A  fourth  is  28 ;  2800. 

Multiply  116  by  25.     A  fourth  is  29;  2900. 

25.  Multiply  22  by  25.  One  fourth  is  5  and  a  half; 
100  times  this  are  5  hundred  and  half  a  hundred ;•  550. 

Multiply  26  by  25.     One  fourth  is  6Jj  650. 

Multiply  28  by  25.     One  fourth  is  7;  700. 

Multiply  30  by  25;  32  by  25;  34  by  25;  36  by  25; 
40  by  25. 

Multiply  42  by  25;  44  by  25;  46  by  25;  48  by 
25;  50  by  25. 

26.  Multiply  13  by  25.  One  fourth  is  3  and  one 
fourth ;  one  hundred  times  this  is  300  and  one  fourth 
of  a  hundred  or  25 ;  325. 

Multiply  15  by  25.  One  fourth  is  3  and  three 
fourths ;  one  hundred  times  this  are  3  hundred  and 
3  fourths  of  a  hundred  or  75;  375. 

Multiply  17  by  25;  19  by  25;  21  by  25;  23  by  25; 
27  by  25  ;  29  by  25';  31  by  25  ;  33  by  25  ;  35  by  25. 

27.  Multiply  116  by  25.     One  fourth  is  29;  2900. 
Multiply  117  by  25.     One  fourth  is  29±;  2925. 
Multiply  121  by  25;  87  by  25;  156  by  25;  960  by  25. 
28.*  What  is  one  third  of  60?    Of  90?    Of  120? 

Of  15?    Of  150?    Of  45?    Of  450? 

What  is  one  third  of  18?  Of  180?  Of  21?  210? 
Of  36?    Of  360?    Of  30?    Of  390? 

What  is  one  third  of  72  ?    Divide  into  60  and  12. 

What  is  one  third  of  54  ?    Of  85  ?    Of  98  ? 

What  is  one  sixth  of  60  ?  Of  80  ?  Divide  into  60 
and  20.  — Of  74?    Of  84?    Of  96?    Of  100? 


*  Note  4. 


20  MENTAL    ARITHMETIC. 

What  is  one  sixth  of  12?  Of  120?  Of  130?  Of 
140  >;  Of  144? 

What  is  one  sixth  of  18  ?5  Of  180?  Of  200?  Of 
210?    Of  220? 

What  is  one  sixth  of  384  ?  Of  492  ?  Divide  into 
480  and  12.  — Of  5551  Divide  into  540  and  15.-^ 
Of  620?    Of  726?    Of  947? 

29.  What  are  the  two  factors  of  18?    Of  180? 
What  are  the  two  factors  of  27  ?    Of  270  ?    Of  22  ? 

Of  220?  Of  35?  Of  350?  Of  54?  Of  540?  Of 
45?  Of  450?  Of  21?  Of  210?  Of  28?  Of  280? 
Of  42?    Of  420? 

30.  What  two  numbers,  multiplied  together,  will 
produce  24? 

What  other  two  factors  will  produce  24?  What 
other  two? 

What  two  factors  will  produce  240?  What  other 
two?     What  others? 

What  two  factors  will  produce  30?    What  others? 

What  two  factors  will  produce  300?    What  others? 

What  two  factors  will  produce  18?    What  others? 

What  two  will  produce  180?     What  others? 

Name  all  the  pairs  of  factors  that  will  produce  36  ; 
360;  48;  480;  60;  600;  64;  640;  72;  720. 

31.  What  is  one  ninth  of  27?  A  man  divided  270 
dollars  equally  among  9  persons.  How  much  did  he 
give  to  each? 

32.  What  is£pne  fourth  of  48  ?  If  480  dollars  are 
divided  into  4  equal  shares,  what  will  each  share  be  ? 

What  is  one  eighth  of  480?  One  sixth  of  480? 
One  twelfth  of  480? 

33.  What  is  one  seventh  of  63  ?  If  a  ship  sails,  at 
a  uniform  rate,  630  miles  in  a  week,  how  many  miles 
does  she  sail  in  a  day  ? 

What  is  one  ninth  of  630?  What  is  one  sixth  of 
630  ?     What  is  one  third  of  630  ? 

34.  What  is   one  fifth   of  25?    If  250   trees   are 


DIVISION.  Xf       27 

-    /   I  I — • 

placed  in  5  equal  rows,  how  many  will  there  be  in 
each  row? 

If  placed  in  50  equal  rows,  how  many  would  there 
be  in  each  row? 

35.  What  is  one  fourth  of  36?  In  a  circle  there 
are  360  degrees.  How  many  are  there  in  one  fourth  of 
a  circle  ?  How  many  in  one  eighth  of  a  circle  ?  How 
many  in  one  sixteenth  of  a  circle? 

36.  What  is  one  eighth  of  56  ?  If  560  trees  were 
planted  in  8  equal  rows,  how  many  would  there  be  in 
each  row?  If  planted  in  16  rows,  how  many  would 
there  be  in  each  row  ? 

37.  What  is  one  eleventh  of  551  If  you  place  550 
trees  in  11  equal  rows,  how  many  will  there  be  in  a 
row  ?  If  you  place  them  in  50  rows,  how  many  will 
there  be  in  each  row  ?  If  you  place  them  in  25  rows, 
how  many  will  there  be  in  each  row  ? 

38.  What  is  one  twelfth  of  96  ?  If  a  man  spends 
960  dollars  in  a  year,  how  much  will  be  his  average 
expense  for  each  month? 

39.  What  is  one  tenth  of  40?    Of  400?    Of  4000? 
What  is  one  fourth  of  40?    400?    Of  4000?    Of 

80?    Of  800?    Of  8000?    One  fourth  of  12  ?    Of  120? 
Of  1200?    Of  12,000? 

40.  What  is  one  fifteenth  of  60?  Of  600?  Of 
6000? 

What  is  one  thirtieth  of  60?  Of  600?  Of  6000? 
Of  1200?    Of  12,000? 

41.  What  is  one  fifth  of  92?  What  is  one  third  of 
51?  One  fourth  of  65?  One  fifth  of  78?  One  sixth 
of  96?  One  seventh  of  100?  Divide  into  70  and  30. 
— What  is  one  ninth  of  117?  Ans.  One  ninth  of  90 
is  10 ;  one  ninth  of  27  is  3 ;  10  and  3  are  13. 

42.  What  is  one  third  of  49?  One  sixth  of  84? 
One  fifth  of  79  ?  One  eighth  of  100  ?  One  seventh  of 
91?  One  sixth  of  79?  One  fourth  of  76?  Of  92? 
Of  57?    Of  60?    Of  52?    Of  65  ?    Of  70? 


28 


MENTAL    ARITHMETIC. 


43.  What  is  one  fourth  of  480?  What  is  one  fifth 
of  155?  Divide  into  150  and  5. — What  is  one  fourth 
of  920?  What  is  one  fifth  of  15,765?  This  number 
may  be  divided  into  15,000,  750,  and  15;  or  15,000, 
500,  250,  and  15. 

44.*  What  is  one  sixth  of  4836?  One  eighth  of 
336?  Divide  into  320  and  16. — What  is  one  seventh 
of  574?  One  third  of  684?  One  "sixth  of  43,248? 
One  ninth  of  72, 108  ?  Of  64,827  ?  One  fifth  of  5275  ? 
One  fourth  of  92,648? 

45.  What  is  one  third  of  6156?    Of  8436? 

46.  What  is  one  fourth  of  6428?    Of  9648?!  .\  I  t 

47.  What  is  one  fifth  of  7655?    Of  12,535? 

48.  What  is  one  sixth  of  13,218?    Of  1944? 

49.  What  is  one  seventh  of  10,542?    Of  14,280? 

50.  What  is  one  eighth  of  1632?    Of  2560? 


SECTION    V. 


TABLE   OF   TIME. 


60  seconds,  (sec.)  .  make  .  1  minute,  .  marked  .  m. 

60  minutes, 1  hour, h. 

24  hours, 1  day, d. 

7  days, 1  week, w. 

4  weeks, 1  month, mo. 

52  weeks,  1  day,  6  hours, .  1  year, y. 

365  days,  6  hours, 1  year, y. 

12  calendar  months, 1  year, y. 

In  common  reckoning,  4  weeks  are  called  a  month ; 
but  this  is  merely  for  convenience  in  doing  business. 

*  Note  5. 


time.  29 

The  number  of  days  in  a  calendar  month  is  30  or  31 ; 
except  February,  which  has  28  days,  and  in  leap  year 
29.  The  6  hours  over  and  above  the  365  days  in  a 
year,  will  in  4  years  amount  to  a  whole  day ;  it  is  then 
added  to  February,  making  29  days,  and  that  year  is 
called  leap  year.  The  number  of  days  in  the  other 
months  may  be  seen  in  the  line  below.  The  months 
connected  by  a  tie  drawn  over  the  words  have  31 
days ;   those  connected  by  a  tie  underneath  have  30. 


Jan.  Feb.  March.  April.  May.  June.  July.  August.  Sept.  Oct.  Nov.  Dec. 
28.  29.  s f  * ' 

You  observe  that,  beginning  with  January,  every 
alternate  month  has  31  days,  till  you  come  to  July  and 
August.  Here  there  are  two  months  together  that 
have  31,  and  then  the  alternation  goes  on  as  before  to 
the  end  of  the  year. 

The  leap  year  may  be  easily  known  from  the  fact 
that  the  number  of  the  year  is  exactly  divisible  by  4. 
Thus  1844  was  leap  year ;  the  number  can  be  divided 
by  4. 

What  years  in  the  present  century  have  been  leap 
years?  What  years  will  be  leap  years  from  now  to 
the  close  of  the  century? 

1.  In  one  minute  there  are  60  seconds ;  in  one  hour 
there  are  60  minutes.  How  many  seconds  are  there 
in  one  hour?  How  many  in  10  hours?  How  many 
in  20  hours?  How  many  in  one  day?  How  many 
in  seven  days,  or  one  week  ?  How  many  in  ten  days  ? 
In  100  days?  In  300  days?  In  350  days?  In  365 
days? 

2.  If  you  save  30  minutes  from  idleness  each  day, 
how  many  hours  will  you  save  in  a  week?  How 
many  in  5  weeks  ?  How  many  in  50  weeks  ?  How 
many  in  52  weeks? 

3.  If  you    read    40    pages   each   day,    how    many 

3  * 


30  MENTAL    ARITHMETIC. 

pages  will  you  read  in  one  week  ?     How  many  in  10 
weeks  ?     How  many  in  52  weeks  ? 

4.  If  a  printer  sets  4  pages  of  type  in  a  day,  in  how 
many  days  will  he  set  the  type  for  a  book  of  500 
pages?  What  will  his  wages  come  to,  at  $1.50  a 
day? 

5.  If  there  are  300  members  in  the  Legislature  of 
Massachusetts,  and  each  member  receives  2  dollars  a 
day  during  the  session,  what  does  the  pay  of  all  the 
members  come  to  for  one  day  ?  What  does  the  pay  of 
the  Legislature  amount  to  for  one  week  ?  For  10 
weeks? 

6.  The  number  of  members  in  Congress  is  about 
275.  At  8  dollars  a  day,  what  is 'the  amount  of  their 
pay  each  day  ?  What  would  be  the  amount  of  their 
pay  for  10  days  ?     For  100  days  ? 

7.  How  many  days  are  there  in  the  3  months  of 
spring  ?  How  many  days  in  the  3  months  of  sum- 
mer ?    jHow  many  days  in  autumn  ? 

8.  How  many  days  in  the  winter  of  leap  year? 
How  many  days  were  there  in  the  winter  of  1844  ? 
How  many  days  in  the  winter  of  1845? 

9.  If  January  comes  in  on  Monday^  on  what  day 
of  the  week  will  February  come  in  ?       * 

If  March  comes  in  on  Wednesday,  on  what  day  of 
the  week  will  April  come  in  ? 

If  August  comes  in  on  Saturday,  on  what  day  of 
the  week  will  September  come  in  ? 

10.  If  April  comes  in  on  Sunday,  on  what  day  of 
the  week  will  it  go  out  ? 

If  June  comes  in  on  Tuesday,  on  what  day  will  it 
go  out  ?  m  . 

If  September  comes  in  on  Saturday,  on  what  day 
will  it  go  out  ? 

11.  If  January  comes  in  on  Friday,  how  many  Sun- 
days will  there  be  in  that  month  ? 


REDUCTION    OF    LINEAR    MEASURE.  31 

If  it  comes  in  on  Thursday,  how  many  Sundays 
will  there  be  in  that  month  ? 

If  June  comes  in  on  Friday,  how  many  Sundays 
will  there  be  in  that  month  ?  If  it  comes  in  on  Sat- 
urday, how  many  ? 

If  February  comes  in  on  Saturday,  and  that  year  is 
leap  year,  hoW  many  Saturdays  will  there  be  in  the 
month  ?     If  it  is  not  leap  year,  how  many  ? 

In  1845,  February  came  in  on  Saturday.  How  many 
Saturdays  were  there  in  that  month  ? 

In  1844,  February  came  in  on  Thursday.  How  many 
Thursdays  were  there  in  that  month  ? 

12.  If  January  comes  in  on  Monday,  on  what  day 
of  the  week  will  March  come  in,  if  it  is  leap  year  ? 
On  what  day,  if  it  is  not  leap  year  ? 

13.  If  June  comes  in  on  Wednesday,  what  day  of 
the  week  will  the  1st  of  August  be?  The  9th?  The 
12th?    The  15th? 


TABLE   OF   LINEAR  MEASURE. 

12    inches,  (in.)  .  make  .  1  foot,.  .  .  marked  ...  ft. 

3  'feet, 1  yard, yd. 

5f  yards,  16£  feet, 1  rod, rd. 

40    rods,  . 1  furlong,  .  ; fur. 

8    furlongs  =  320  rods,  .  1  mile, m. 

3    miles, 1  league, 1. 

694-  miles, 1  degree  of  latitude,  .  .  deg. 

For  lengths  less  than  an  inch,  the  inch  is  divided 
into  fourths,  eighths,  tenths,  or  twelfths. 

I.  How  many  inches  in  2  feet?  In  4  feet?  In  5 
feet  ?  In  7  feet  ?  In  10  feet  ?  In  12  feet  ?  How 
many  inches  in  4  yards  ?  In  1  rod  ?  In  3  rods  ?  How 
many  feet  in  1  furlong  ?  In  2  furlongs  ?  In  4  fur- 
longs ?     In  1  mile  ? 


32  MENTAL    ARITHMETIC. 

2.  How  many  miles  in  46  leagues  ?  In  132  leagues  ? 
How  many  miles  in  2  degrees  of  latitude  ?  In  3  de- 
grees?    In  4£  degrees  ?     In  6  degrees? 

In  estimating  the  miles  in  any  number  of  degrees 
of  latitude,  it  is  most  convenient  to  call  a  degree  70 
miles ;  and  then,  if  we  wish  to  be  accurate,  we  may 
subtract  from  the  answer  half  as  many  miles  as  there 
are  degrees.  In  this  way,  the  distance  of  places  from 
each  other  may  be  determined  on  a  map.  The  degrees 
of  latitude  on  the  margin  may  be  used  as  a  scale  of 
miles.  If  the  distance  of  two  places  from  each  other 
is  equal  to  6£  degrees  of  latitude,  how  many  miles  are 
they  apart  ? 

3.  How  many  yards  in  10  rods  ?  In  20  rods  ?  In 
30  rods  ?     In  1  furlong  ?     In  8  furlongs,  or  1  mile  ? 

4.  In  measuring  land  or  a  road  with  a  chain  4  rods 
long,  how  many  times  must  the  chain  be  applied  to 
the  ground  in  measuring  one  mile  ?  How  many  times 
in  measuring  the  road  from  Boston  to  Salem,  15  miles? 
How  many  in  measuring  from  Boston  to  Providence, 
40  miles? 

5.  If  a  man  walks  3  miles  in  an  hour,  how  many 
minutes  will  he  be  in  walking  1  mile  ?  How  many 
minutes  in  walking  1  fourth  of  a  mile?  How  many 
rods  will  he  walk  in  1  minute  ?     Ans.  16. 

How  many  seconds  will  he  be,  then,  in  walking  1 
rod?  16  will  go  into  60,  3  times  and  12  over.  He 
will  be,  then,  a  little  less  than  4  seconds  in  walking  1 
rod. 

Let  us  now  suppose  he  is  precisely  4  seconds  in 
walking  1  rod  ;  how  many  rods  would  he  walk  in  a 
minute  ?  How  many  in  10  minutes  ?  How  many 
in  60  minutes?     How  many  miles  ? 

6.  If  a  man  in  walking  takes  6  steps  to  a  rod,  how 
many  steps  will  he  take  in  walking  a  mile  ?  How 
many  in  walking  10  miles  ?  How  many  in  walking 
40  miles  ? 


REDUCTION    OF    LINEAR    MEASURE,  33 

7.  If  a  man  in  walking  takes  6  steps  to  a  rod,  and 
takes  2  steps  in  a  second,  how  many  seconds  will  he 
be  in  walking  one  rod  ?  How  many  seconds  in  walk- 
ing 10  rods  ?  20  rods  ?  If  a  man  walks  20  rods  in 
one  minute,  how  many  minutes  will  it  take  him  to 
walk  a  mile  ?  20  are  contained  in  320  just  as  many 
times  as  2  are  contained  in  32. 

8.  If  a  man  walks  20  rods  in  one  minute,  how  long 
will  it  take  him  to  walk  4  miles  ? 

9.  If  a  railroad  train  goes  30  miles  in  an  hour,  how 
far  does  it  go  in  one  minute  ?  How  many  rods  in  one 
second  ? 

Ans.  30  miles  in  60  minutes  is  1  mile  in  2  minutes  ; 
half  a  mile  in  one  minute  ;  quarter  of  a  mile  in  half 
a  minute  ;  that  is  80  rods  in  30  seconds  ;  that  is  8  rods 
in  3  seconds  ;  and  in  1  second,  one  third  of  8  rods,  or 
2  rods  and  two  thirds. 

10.  How  many  rods  in  14  miles  ? 

In  1  rod  there  are  16£  feet.  In  1  mile  there  are  320 
rods.  How  many  feet  are  there  in  a  mile  ?  There 
are  various  ways  of  finding  the  answer  to  this  ques- 
tion ;  some  of  them  will  be  suggested,  and  the  pupil 
left  to  take  his  choice. 

First,  how  many  feet  are  there  in  300  rods  ? 

This  is  not  difficult ;  for  in  3  rods  there  are  three 
times  16^  feet,  and  in  300  rods  there  are  100  times  as 
many.  3  times  15  feet  are  45  feet ;  3  times  l£  feet 
are  4|,  which  added  to  45  make  49£  feet  in  3  rods. 
Now,  100  times  49  are  4900,  and  100  halves  are  50  ; 
4950  feet  in  300  rods.  In  20  rods  there  are  ten  times 
as  many  feet  as  in  2  rods  ;  in  2  rods  there  are  twice 
16£  or  33;  in  20  rods,  therefore,  there  are  330  feet; 
300  added  to  4900  make  5200,  and  30  added  to  50 
make  80  ;  there  are,  then,  5280  feet  in  a  mile. 

Another  method  would  be  to  multiply  320  first  by 
8,  and  that  product  by  2,  for  8  and  2  are  the  factors 
of  16 ;.  then,  as  there  was  £  a  foot  in  each  rod  left  out, 

C 


34  MENTAL    ARITHMETIC. 

there  must  be  added  half  as  many  feet  as  there  are 
rods,  or  half  of  320. 

Another  method  would  be  to  multiply  320  by  10, 
then  by  6,  and  add  the  products,  and  lastly  by  J-,  and 
add  that  to  the  other  products. 

The  pupil  can  try  each  of  these  ways,  and  see  if  he 
obtains  the  same  answer. 

Let  us  now  see  if  our  answer  is  correct.  If  there 
are  5280  feet  in  a  mile,  how  many  are  there  in  half  a 
mile  ?  One  half  of  5200  is  2600  ;  one  half  of  80  is  40 ; 
there  are,  then,  2640  feet  in  half  a  mile.  How  many 
in  1  fourth  of  a  mile?  One  half  of  2640  feet,  which 
is  1320.  Now,  1  fourth  of  a  mile  is  80  rods.  If,  then, 
there  are  1320  feet  in  80  rods,  how  many  will  there 
be  in  8  rods  ?  One  tenth  as  many.  One  tenth  oY  1320 
is  132.  Now,  how  many  are  there  in  one  rod  ?  One 
eighth  of  132 ;  dividing  132  into  80  and  52 ;  one  eighth 
of  80  is  10,  and  one  eighth  of  52  is  6  J  or  4-,  which  added 
to  10  make  164.  We  have  now  come  down  from  5280, 
and  arrived  by  successive  divisions  to  164,  the  number 
from  which  we  started  at  first.  The  answer  is  thus 
proved  to  be  correct. 

11.  How  many  feet  are  there  in  2  rods?  In  20 
rods  ?     In  200  rods  ? 

12.  How  many  feet  are  there  in  3  rods?  In  30 
rods  ?  In  300  rods  ?  In  8  rods  ?  In  80  rods,  or  a 
quarter  of  a  mile  ? 

13.  How  many  rods  are  there  in  2  miles?  In  4 
miles?  In  8  miles?  In  20  miles?  In  30  miles?  In 
50  miles  ? 

14.  How  many  rods  are  there  in  half  a  mile  ?  In 
three  fourths  of  a  mile  ?  In  1  mile  and  a  half?  In  1 
mile  and  3  furlongs  ?  In  2  miles  and  5  furlongs  ?  In 
4  miles  and  7  furlongs  ? 

15.  How  many  yards  are  there  in  2  rods  ?  In  20 
rods  ?     In  3  rods  ?     In  30  rods  ?     In  300  rods  ? 


REDUCTION    OF    LINEAR    MEASURE.  35 

How  many  yards  are  there  in  3  rods  and  4  feet  ? 
How  many  yards  are  there  in  17  rods  and  11  feet? 

16.  How  many  inches  are  there  in  7  feet  ?  In  9 
feet?  In  6  feet?  In  8  feet  and  6  inches?  In  11  feet 
9  inches  ? 

17.  How  many  inches  are  there  in  1  rod,  or  16£  feet  ? 
How  many  inches  in  2  rods  ?    In  3  rods  ?     In  4  rods  ? 

18.  A  house  is  46  feet  and  5  inches  in  length. 
How  many  inches  long  is  it? 

A  creeping  vine  grows  on  an  average  3  inches  a 
day.  How  many  days  will  it  take  to  grow  from  the 
ground  to  the  top  of  a  house  that  is  25  feet  high  ? 

19.  A  stage-horse  travels  13  miles  and  20  rods  each 
day.  How  far  will  he  travel  in  60  days  ?  How  far 
in  120? 

20.  If  a  horse  travels  16  miles  in  a  day,  how  many 
miles  will  he  travel  in  5  days?  In  10  days?  In  15 
days  ?     In  20  days  ?     In  200  days  ?     In  300  days  ? 

21.  What  is  the  weight  of  iron  used  in  one  mile  of 
railroad,  allowing  50  pounds  for  a  yard  of  rail  ? 

If  one  yard  of  rail  weighs  50  pounds,  the  two  rails 
for  one  yard  of  the  road  would  weigh  100  pounds:  from 
this  may  be  obtained  the  weight  for  one  rod ;  for  10 
rods;  for  100  rods;  for  300  rods;  for  320  rods. 

22.  If  a  yard  of  iron  rail  weighs  75  pounds,  what 
would  be  the  weight  of  the  two  rails  of  a  single  track 
for  one  rod  ?  For  2  rods  ?  For  3  rods  ?  For  300 
rods  ?     For  20  rods  ?     For  200  rods  ? 

What  would  be  the  cost  of  the  rail  for  one  rod,  at  2 
cents  a  pound?  At  3  cents  a  pound?  At  4  cents  a 
pound  ? 

23.  If  the  cost  of  the  iron  for  a  single  track  of  railroad 
is  6000  dollars  a  mile,  and  the  cost  of  the  land  and  the 
labor  of  construction  equals  that  of  the  iron,  what 
would  be  the  cost  of  15  miles  of  railroad  ?    Of  24  miles  ? 

24.  What  would  be  the  cost  of  constructing  1  mile  of 
common  road  at  225  dollars  a  rod  ? 


36  MENTAL    ARITHMETIC. 

25.  What  would  be  the  cost  of  building  80  rods  of 
common  wall  at  54  cents  a  rod  ? 

26.  If  a  horse  travels  10  miles  in  an  hour,  how  long 
is  he  in  travelling  1  mile  ?  How  long  in  travelling  £ 
of  a  mile,  or  80  rods?  How  many  seconds  is  he  in 
travelling  8  rods  ?    How  long  in  travelling  1  rod  ? 

27.  A  body  falling  through  the  air,  falls  in  the  first 
second  16£  feet,  and  in  each  succeeding  second  it  falls 
twice  16^  feet  farther  than  in  the  preceding  second. 
How  far  would  a  stone  fall  in  2  seconds  ? 

28.  How  far  would  it  fall  in  the  third  second? 
How  far  would  it  fall  in  3  seconds? 

29.  How  far  would  it  fall  in  the  fourth  second? 
How  far  would  it  fall  in  4  seconds? 

30.  Sound  moves  through  the  air  at  the  rate  of 
1090  #  feet  in  a  second.  How  many  feet  will  it  move 
in  3  seconds  ?  How  many  feet  in  4  seconds  ?  How 
many  feet  in  5  seconds? 

As  sound  is  found  thus  to  pass  5450  feet  in  5  sec- 
onds, and  as  there  are  5280  feet  in  a  mile,  we  see  that 
in  5  seconds  sound  moves  170  feet  more  than  a  mile. 
Now,  as  165  feet  is  just  10  rods,  we  say,  without 
much  error,  that  sound  moves  1  mile  and  10  rods  in 
5  seconds.  This  is  accurate  enough  for  all  common 
purposes,  and  you  will  do  well  to  fix  it  in  your  mem- 
ory, and  make  your  calculations  from  it. 

31.  How  many  rods  will  sound  move  in  1  second? 
One  fifth  of  320+10  rods  =  66  rods. 

32.  How  many  rods  in  2  seconds?  How  many 
rods  in  4  seconds? 

Thus,  if  you  watch  the  stroke  of  an  axe  used  by 
some  one  at  a  distance,  and  observe  that  the  sound 
comes  to  you  one  second  later  than  you  see  the  stroke, 
you  may  know  that  the  distance  is  66  rods.  If  the 
sound  of  a  bell  comes  to  you  two  seconds  after  the 

*  Professor  Pierce  on  Sounds. 


REDUCTION    OF    LINEAR    MEASURE.  37 

stroke  is  given,  you  must  be  distant  from  the  bell  132 
rods.  In  these  cases,  no  allowance  is  made  for  the 
transmission  of  light.  You  are  supposed  to  see  the 
motion  as  soon  as  it  occurs.  This  is  not  strictly  the 
fact ;  but  the  time  is  so  exceedingly  small,  that  it  need 
not  be  taken  into  the  account. 

33.  In  a  still  night,  a  church  bell  is  sometimes 
heard  at  the  distance  of  12  miles.  How  many  sec- 
onds, or  nearly  how  many,  after  the  stroke,  would 
the  sound  be  heard  at  that  distance? 

34.  If  the  report  accompanying  a  flash  of  lightning 
is  heard  4  seconds  after  the  flash  is  seen,  how  far  from 
the  hearer  was  the  discharge  ?  How  far,  if  the  time 
between  the  flash  and  the  report  is  6  seconds?  How 
far,  if  the  time  is  8  seconds?  How  far,  if  the  time  is 
10  seconds  ?    How  far,  if  the  time  is  15  seconds  ? 

35.  The  report  of  a  cannon  has,  in  some  instances, 
been  heard  at  the  distance  of  100  miles.  Allowing  that 
the  sound  moves  one  mile  in  5  seconds,  in  how  many 
seconds  after  the  discharge  would  the  report  be  heard  at 
the  distance  of  100  miles? 

36.  By  means  of  a  magnetic  telegraph,  it  is  possi- 
ble to  communicate  intelligence  instantly  from  New 
Orleans  to  Boston,  a  distance  of  1500  miles.  If  this 
intelligence  could  be  communicated  by  sound  passing 
through  the  air,  how  long  would  it  be  in  travelling 
that  distance,  allowing  5  seconds  to  a  mile  ? 

A  ball  discharged  from  a  gun  moves  at  first  with  a 
greater  speed  than  sound,  but  it  moves  slower  and 
slower,  and  before  it  is  spent  the  report  overtakes  it, 
and  passes  by  it ;  for  sound  moves  always  at  the  same 
rate. 

37.  If  a  cannon  ball  moves  a  mile  in  8  seconds, 
how  long  would  it  be  in  moving  3  miles  ?  How  long 
in  moving  one  fourth  of  a  mile  ?  How  long  in  moving 
one  eighth  of  a  mile  ?    How  long  in  moving  If  miles  ? 

4 


38  MENTAL    ARITHMETIC. 

SECTION     VI. 

TABLE   OF  FEDERAL   MONEY 

10  mills   .  .  .  make  ...  1  cent, .  .  .  marked .  .  .  ct. 

10  cents, 1  dime, d. 

10  dimes, 1  dollar, D. 

10  dollars, 1  eagle, E. 

This  is  established  by  law  as  the  currency  of 
the  United  States. 

The  general  mark  for  Federal  Money  is  $  ;  as, 
$5.14,  five  dollars  fourteen  cents.  A  period  must 
always  be  placed  between  dollars  and  cents. 

1.  How  many  mills  in  2  cents?  In  10  cents?  In 
12  cents?  In  5£  cents?  In  12^  cents?  In  36  cents? 
In  1  dollar? 

2.  How  many  cents  in  5  dimes  ?  In  1 1  dimes  ?  In 
16  dimes?  In  U  dollars?  In  17 i  dollars?  In  12£ 
dollars  ? 

3.  How  many  dimes  in  7  dollars  ?  In  13i  dollars  ? 
In  3  eagles  ?    In  56  dollars  ?    In  100  dollars  ? 

4.  How  many  cents  in  35  mills?  In  180  mills?  In 
600  mills?  How  many  dimes  in  80  cents?  In  210 
cents?    In  740  cents? 

5.  How  many  dollars  in  350  cents?  In  325  cents? 
In  700  cents?  In  850  cents?  In  1400  cents?  In 
1675  cents?    In  925  cents? 


TABLE   OF   STERLING   MONEY. 

4  farthings,  (qr.)  make  1  penny,  .  .  marked'.  .  d. 

12  pence, 1  shilling, s. 

20  shillings, 1  pound, £. 

This  is  the  currency  of  Great  Britain. 


REDUCTION.  39 

1.  How  many  farthings  are  there  in  3  pence?  In 
7  pence?    In  8  pence?    In  10  pence?    In  11  pence? 

2.  How  many  pence  in  2  shillings  ?  In  12  shillings  ? 
In  15  shillings?    In  18  shillings?    In  16  shillings? 

3.  How  many  shillings  in  4  pounds  ?  In  7  pounds  ? 
In  18  pounds  ?    In  36  pounds  ?    In  84  pounds  ? 

4.  How  many  farthings  in  1  shilling  and  6  pence  ? 
In  2  shillings  and  6  pence?  In  15  shillings  and  4 
pence  ? 

How  many  pence  in  10  shillings?  In  20  shillings? 
In  2  pounds  ?    In  4  pounds  ?    In  12  pounds  ? 

5.  How  many  farthings  in  1  pound  ?  In  5  pounds  ? 
In  8  pounds  ?    In  1  pound  2  shillings  ? 

6.  How  many  pence  in  45  farthings?  In  128  far- 
things? In  464  farthings?  In  1296  farthings?  In 
648  farthings? 

7.  How  many  shillings  in  80  pence  ?  In  67  pence  ? 
In  372  pence  ?    In  649  pence  ?    In  840  pence  ? 

8.  How  many  pounds  in  267  shillings?  In  845 
shillings?  In  432  shillings?  In  640  shillings?  In 
4000  shillings? 

9.  How  many  pounds  in  890  pence?  In  16,000 
farthings?  In  720  pence?  In  1200  pence?  In  456 
pence  ? 

10.  How  many  pence  in  5  pounds  4  shillings  ?  In 
7  pounds  8  shillings  ?    In  12  pounds  3  shillings  ? 

How  many  farthings  in  4  shillings  6  pence?  In  9 
shillings  ?  How  many  farthings  in  6  pounds  3  shil- 
lings 8  pence  ? 

11.  A  man  set  out  on  a  journey  with  £4  8s.  6d.  in 
his  pocket.  Before  spending  any  thing,  he  received 
in  payment  of  a  debt  £2  3  s.  8d.  How  much  had  he 
then  ?  When  he  arrived  home,  he  had  spent  £1  4  s.  6d. 
How  much  had  he  then? 

These  denominations,  you  must  bear  in  mind,  have 
not  the  same  value  in  English  currency  that  they 
have  iii  the  United  States. 


40 


MENTAL    ARITHMETIC. 


In  our  country,  they  have  different  values  in  the 
different  States,  but  in  none  of  them  so  high  a  value 
as  in  England.  In  the  New-England  States,  a  shil- 
ling is  equal  to  16  cents  and  two  thirds,  and  6  shillings 
make  a  dollar.  In  New  York,  12  and  a  half  cents  are 
a  shilling,  and  8  shillings  a  dollar.  In  other  States, 
the  values  are  still  different.  But  these  denominations 
are  gradually  giving  way  to  those  of  the  Federal  cur- 
rency. They  are  now  used  only  in  naming  prices. 
Accounts  are  not  kept  in  them,  and  all  that  is  impor- 
tant in  them  may  be  learned  by  practice  without  fur- 
ther notice  here. 

In  the  Sterling  currency,  used  in  England,  a  pound 
is  equal  to  4  dollars,  44  cents,  and  4  mills ;  10  shillings, 
therefore,  or  half  a  pound,  are  2  dollars,  22  cents,  2 
mills ;  and  1  shilling  is  one  tenth  part  of  that,  or  22 
cents  2  mills.  An  English  sixpence  is,  therefore,  11 
cents  1  mill.  The  following  table  will  be  useful  in 
exchanging  English  money  to  our  own. 

1  -pound  (£ )  is $ 4.44  4 

10  shillings,  or  half  a  pound, 2.22  2 

1  shilling, 22  2 

6  pence,  or  half  a  shilling, Ill 

4  shillings  6  pence, 1.00  0 

1  guinea,  21  shillings, 4.66  6 

The  actual  value  of  the  English  money  is  a  little 
higher  than  is  here  stated,  but  this  is  sufficiently  accu- 
rate for  a  general  table. 

1.  What  is  the  value,  in  dollars  and  cents,  of  2£  ? 
3£?  4£?  5£?  l£6s.?  2£8s.?  3s.  6d.?  5s.  9d.? 


REDUCTION. 


TABLE   OF  DRY  MEASURE. 


41 


2  pints  (pt.)  .  .  make  .  .  1  quart,  .  .  .  marked  .  .  qt. 

4  quarts,  .  , 1  gallon, gal. 

8  quarts, 1  peck, pk. 

4  pecks, 1  bushel, bu. 

8  bushels, 1  quarter, qr. 

3(5  bushels, 1  chaldron, ch. 

These  denominations  are  used  for  measuring  grain, 
fruit,  and  coal.  The  pint,  quart,  and  gallon,  are  larger 
than  the  same  denominations  in  wine  measure,  and 
less  than  those  of  beer  measure. 


1.  How  many  pints  in  1  peck  ?  In  3  pecks  ?  In  1 
bushel  ?     In  3  bushels  ?     In  4  bushels  ? 

2.  How  many  quarts  are  there  in  1  bushel  ?  In  4 
bushels  ?  How  many  pecks  in  7  quarters  ?  In  2 
chaldrons  ? 

3.  If  a  horse  eat  4  quarts  of  oats  each  day,  how 
many  bushels  will  he  eat  in  10  weeks  ?  How  many 
bushels  in  50  weeks  ?     In  52  weeks  ? 

What  will  they  cost  at  50  cents  a  bushel  ? 

4.  In  80  quarts  how  many  pecks?  How  many 
bushels  ? 

In  644  quarts  how  many  pecks  ?  How  many 
bushels? 

In  7840  quarts  how  many  pecks?  How  many 
bushels? 

5.  In  100  pints  how  many  pecks?  How  many 
bushels  ? 

In  620  pints  how  many  pecks?  How  many 
bushels  ? 

4*' 


42  MENTAL    ARITHMETIC. 


TABLE   OF  AVOIRDUPOIS   WEIGHT. 

16  drams  (dr.)  .  .  make  .  .  1  ounce,  .  .  marked  .  .  oz. 

16  ounces, 1  pound, lb. 

25  pounds, 1  quarter,  (net  wt.)  .  .  qr. 

28  pounds,. 1  quarter,  (gross  wt.) .  qr. 

4  quarters, 1  hundred  weight,  .  .  .  cwt. 

20  hundred  weight,   ....  1  ton, T. 

These  denominations  are  used  in  weighing  hay, 
grain,  meat,  flour,  and  all  the  most  common  articles 
bought  and  sold  by  weight.  On  account  of  the  waste 
in  handling  such  articles,  their  shrinking  in  drying, 
and  worthless  admixtures  sometimes  found  in  them, 
112  pounds  are  sometimes  allowed  for  one  hundred 
weight ;  this  makes  28  pounds  one  quarter,  and  is 
called  gross  weight.  In  all  the  following  questions 
of  avoirdupois  weight,  understand  gross  weight,  un- 
less net  weight  is  expressed. 

1.  How  many  drams  in  3  oz.  ?  In  5  oz.  ?  In  8  oz.  ? 
In  11  oz.  ?  How  many  oz.  in  12  lbs.?  In  15  lbs.? 
In  20  lbs.  ?     In  32  lbs.  ?     In  45  lbs.  ? 

2.  How  many  lbs.  in  4  cwt.  net  weight  ?  In  4  cwt. 
gross  ?  In  6  cwt.  net  weight  ?  In  6  cwt.  gross  ?  In 
5  cwt.  2  qrs.  net?  In  5  cwt.  2  qrs.  gross  ?  In  7  cwt. 
3  qrs.  net  weight  ?     In  7  cwt.  3  qrs.  gross  ? 

3.  How  many  lbs.  in  a  ton  net  weight  ?  In  a  ton 
gross  ?  How  many  lbs.  in  5  tons  3  cwt.  net  ?  In  5 
tons  3  cwt.  gross  ? 

4.  There  are  2  loads  of  hay  whose  net  weight  is  as 
follows :  the  first,  25  cwt.  3  qrs.  17  lbs. ;  the  second, 

17  cwt.  2  qrs.  21  lbs.     What  is  the  weight,  of  both  ? 

5.  A  man  set  out  for  market  with  a  load  of  hay 
weighing  36  cwt.  2  qrs.  15  lbs.  net  weight.      He  lest 


REDUCTION.  43 

a  part  of  it ;  the  remainder  weighed  25  cwt.  1  qr.  8 
lbs.     How  much  did  he  lose  ? 

6.  If  there  are  196  lbs.  in  a  barrel  of  flour,  how 
many  lbs.  net  weight  are  there  in  10  barrels  ? 

196  lbs.  are  7  quarters  gross.  How  many  cwt.  gross 
are  there  in  10  barrels  of  flour  ? 

7.  How  many  pounds  are  there  in  100  oz.  ?  In 
650  oz.  ? 

8.  A  barrel  of  flour  weighs  7  quarters  gross.  How 
many  tons  gross  are  there  in  100  barrels  of  flour? 

9.  What  will  be  the  expense  of  transporting  by 
railroad  100  barrels  of  flour,  100  miles,  at  the  rate 
of  3  dollars  a  ton  ? 

What  will  be  the  expense  of  transporting  a  single 
barrel  ? 

100  barrels  are  700  qrs.  gross  weight.  400  qrs.  = 
100  cwt.  =  5  tons;  300  qrs.  =  75  cwt.  =  3  fons  15 
cwt.  ;  this,  added  to  5  tons,  makes  8  tons  15  cwt. 

10.  The  freight  of  goods  by  wagon  is  about  20  dol- 
lars a  ton  gross  for  100  miles.  .At  this  rate,  what  will 
be  the  cost  of  carrying  a  barrel  of  flour  100  miles  ? 


TABLE   OF   TROY   WEIGHT. 

24  grains  (gr.)  .  make  .  1  pennyweight,  marked  dwt. 

20  pennyweights, 1  ounce, oz. 

12  ounces, 1  pound, lb. 

This  is  used  for  weighing  gold  and  silver.  The 
pound  Troy  is  nearly  one  fifth  less  than  the  pound 
Avoirdupois. 

1.  How   many  grains    in  6  pennyweights?     In   8 
pennyweights  ?     In  12  pennyweights  ?     In  1  oz.  ?     In 
/  2  oz.  ?     In  4  oz.  ?     In  6  oz.  ? 


44  MENTAL    ARITHMETIC. 

2.  How  many  pennyweights  in  8  oz.  ?  In  11  oz.  ? 
In  1  lb.  ?  In  3  lbs.  ?  In  8  lbs.  ?  In  5  lbs.  ?  In  1  lb. 
3  oz.?     In  2  lbs.  5  oz.  ? 

3.  How  many  oz.  in  120  dwt.  ?  In  480  dwt.  ?  In 
060  grs.  ?  How  many  lbs.  in  100  oz.  ?  In  860  dwt.  ? 
In  1200  dwt.  ? 


TABLE   OF   APOTHECARIES'  WEIGHT.  | 

20  grs make 1  scruple, .  .  .  marked ...  9 

3  scruples, 1  dram, 5 

8  drams, 1  ounce,   .  .  .  . , § 

12  ounces, 1  pound, lb 

This  table  is  used  only  by  apothecaries  in  mixing 
medicines.  The  pound  and  ounce  are  the  same  as  in 
Troy  weight. 

TABLE   OF   CLOTH   MEASURE. 

2£ inches   (in.)    .  make    .  1  nail,  .  .  .  marked  .  .  .  na. 
4   nails, 1  quarter, qr. 

4  quarters, 1  yard, yd. 

3   quarters,  . 1  ell  Flemish, Fl.  e. 

5  quarters, 1  ell  English, E.  e. 

6  quarters, 1  ell  French, Fr.  e. 

1.  How  many  inches  in  1  q*  ?  In  1  yd.  ?  In  3 
yds.  ?     In  I  ell  Eng.  ?     In  1  ell  Fr.  ?     In  1  ell  Fl.  ? 

2.  How  many  inches  in  4  yds.  ?  In  7  yds.  ?  In  12 
yds.?     In  10  yds.?     In  20  yds.?     In  6  yds.  3  qrs.? 

? 


In  4  yds.  1  qr.  ? 


REDUCTION.  45 


TABLE   OF   WINE  MEASURE. 

4  gills  (gi.)  .  .  make  .  .  1  pint,  .  .  .marked.  .  .  pt. 

2  pints, 1  quart, qt. 

4  quarts, 1  gallon, gal. 

3H  gallons, 1  barrel, bl. 

63  gallons, 1  hogshead, hhd. 

2  hogsheads, 1  pipe, -p. 

2  pipes, 1  tun, T. 

This  table  is  used  for  measuring  wine,  spirits,  cider, 
and  water. 

1.  How  many  gills  in  1  quart  ?  In  1  gal.  ?  In  4 
gals.  ?  In  6  gals.  ?  In  10  gals.  ?  In  13  gals.  ?  In 
15  gals.  ? 

2.  How  many  pints  in  1  gal.  ?  In  4  gals.  ?  In  20 
gals.  ?     How  many  qts.  in  1  barrel  ?     In  1  hogshead  ? 

3.  How  many  gallons  in  5  barrels?  In  8  barrels? 
How  many  gals,  in  half  a  barrel  ?  In  one  fourth  of  a 
barrel  ? 

4.  In  100  gals,  how  many  barrels?  In  300  gals, 
how  many  bis.  ? 

5.  At  14  cents  a  gallon,  what  is  1  qt.  of  vinegar 
worth  ?  3  qts.  ?  6  qts.  ?  10  qts.  ?  15  qts.  ?  21  qts.  ? 
30  qts.  ? 

6.  What  is  1  barrel  of  vinegar  worth  at  15  cts.  a 
gallon  ?     How  much,  if  the  price  is  20  cts.  a  gal.  ? 

TABLE  OF  ALE  OR  BEER  MEASURE. 

(Used  in  measuring  malt  liquors  and  milk.) 

2  pints"  (pt.)  .  .  make  .  .  1  quart,  .  .  .marked.  .  .  qt. 

4  quarts, 1  gallon, gal. 

36  gallons, 1  barrel, bl. 

The  beer  gallon  is  a  little  more  than  one  fifth  larger 


46  MENTAL    ARITHMETIC. 

than  the  wine  gallon.  There  are  other  measures  of 
beer  besides  those  in  the  tables  ;  as,  the  firkin,  of  9 
gallons ;  the  kilderkin,  18  ;  the  hogshead,  54  •  but 
these  are  not  much  used  in  this  country.  A  barrel  of 
wine  contains  not  quite  three  fourths  as  much  as  a 
barrel  of  beer. 

1.  In  1  bl.  how  many  pints  ?  How  many  pints  in 
3  bis.  ?  How  many  gallons  in  5  bis.  ?  In  12  bis.  ? 
In  15  bis.  ?     In  21  bis.? 

2.  In  100  gallons  how  many  bis.  ?  How  many 
bis.  in  400  gals.  ?  First  consider  how  many  bis.  there 
are  in  360  gals. 


MEASURE   OF   THE    CIRCLE. 

Every  circle  is  supposed  to  have  its  circumference 
divided  into  360  equal  parts,  called  degrees  ;  and  each 
degree  into  60  parts,  called  minutes  ;  and  each  minute 
into  60  pjarts,  called  seconds.  Whether  the  circle  is 
great  or  small,  it  is  still  divided  into  360  degrees.  A 
degree,  therefore,  is  always  the  same  fixed  part  of  the 
circumference  of  a  circle,  although  its  actual  length  is 
longer  or  shorter,  according  as  the  circle  is  great  or 
small.  The  line  passing  from  the  centre  to  the  cir- 
cumference is  called  the  radius  of  the  circle.  To 
give  you  some  idea  of  the  length  of  a  degree  in  circles 
of  different  magnitudes,  I  will  state  that,  on  comparing 
a  degree  in  any  circle  with  its  radius,  it  has  been 
found  to  be  about  one  fifty-eighth  part  of  it.  In  other 
words,  58  degrees  on  the  circumference  of  a  circle  are 
about  equal  to  the  radius.  If  a  degree  is  1  inch,  the 
radius  of  that  circle  is  58  inches.  If  the  radius  of  a 
carriage  wheel  is  29  inches,  a  degree  on  the  rim  of  the 
same  wheel  will  be  half  an  inch. 

If  we  take  for  illustration  one  of  the  largest-sized 
water-wheels,  29  feet  in  diameter,  a  degree  on  its  rim 
would  measure  only  3  inches. 


MEASURE    OF    THE    CIRCLE.  47 

You  may  enlarge  the  circle  in  your  mind,  till  you 
suppose  it  extending  over  a  plain,  with  a  radius  of  58 
rods.  A  degree  on  such  a  circle  will  measure  1  rod. 
If  the  radius  is  58  miles,  a  degree  will  measure  1  mile. 
Now,  the  circle  round  the  earth  is  so  great  in  extent 
that  a  degree  measures  69£  miles.  This  may  aid  you 
in  forming  a  conception  of  the  vast  magnitude  of  the 
earth. 

Each  of  these  degrees  is  divided  into  60  minutes,  or 
geographical  miles.  A  geographical  mile,  therefore,  is 
about  one  sixth  greater  than  a  common  mile.  The 
Table  of  Circular  Measure  is  as  follows  : 

60  seconds  (/7).  .  .  make.  .  .  1  minute,  .  .  marked  .  '. 

60  minutes,  (or  geog.  miles,)  1  degree, °. 

360  degrees a  circle. 

The  term  miles,  instead  of  minutes,  can  be  used 
only  in  reference  to  the  great  circle  of  the  earth. 

As  the  earth  turns  round  on  its  axis  once  in  24 
hours,  every  place  upon  it  passes  in  that  time  through 
the  360  degrees  of  its  circle ;  and  on  the  equator, 
which  is  the  great  circle,  each  of  these  degrees,  we 
have  seen,  is  69£  miles. 

How  swiftly,  then,  does  a  body  lying  on  the  equator 
move  in  consequence  of  the  daily  revolution  of  the 
earth  ? 

In  24  hours,  it  passes  through  360  degrees  ;  in  one 
hour,  then,  it  will  pass  through  one  twenty-fourth 
part  as  many,  which  is  15  degrees.  If  it  pass  through 
15  degrees  in  one  hour,  how  many  minutes  will  it  be 
in  passing  through  1  degree  ?  One  fifteenth  of  60 
minutes  is  4  minutes.  If  it  pass  through  a  degree  in 
4  minutes,  what  part  of  a  degree  will  it  pass  through 
in  1  minute  ?  One  fourth  of  a  degree,  or  15  geo- 
graphical miles.  If  it  pass  through  15  geographical 
miles  in  1  minute,  in  how  many  seconds  will  it  pass 


48  MENTAL    ARITHMETIC. 

through  one  geographical  mile  ?  In  4  seconds  ;  and 
in  1  second  it  will  pass  through  one  fourth  of  a  geo- 
graphical mile. 

Now,  a  geographical  mile  on  the  equator  is,  as  we 
have  seen,  longer  than  a  common  mile.  We  will  here 
suppose  it  no  longer,  but  of  the  same  length,  and  it 
appears  that  an  object  on  the  equator  moves,  as  the 
vast  earth  whirls  round  on  its  axis,  one  quarter  of  a 
mile  every  second  of  time.  Reflect  now,  that,  while 
the  surface  of  the  earth  moves  with  such  amazing 
speed,  so  vast  is  its  size,  that  it  occupies  an  entire  day 
and  night  in  turning  once  round. 

If,  as  above  stated,  the  earth  turns  from  west  to  east 
at  the  rate  of  15  degrees  in  an  hour,  we  can,  by 
knowing  the  time  of  day  in  any  place,  ascertain  what 
time  it  is  at  a  place  any  particular  number  of  degrees 
east  or  west  of  it.  It  is  noon  at  any  place  when  the 
meridian  of  that  place  passes  under  the  sun. 

1.  When  it  is  noon  at  Boston,  what  time  is  it  at  a 
place  15  degrees  west  of  Boston  ?  At  a  place  15  de- 
grees east  of  Boston  ? 

2.  When  it  is  12  o'clock  at  Boston,  what  time  is  it 
at  a  place  1  degree  west  of  Boston  ?  At  a  place  1 
degree  east  of  Boston  ?  At  a  place  2  degrees  west  of 
Boston  ?  At  a  place  2  degrees  east  of  Boston  ?  3 
degrees  east  ?  3  degrees  west  ?  4  degrees  east  ?  4 
degrees  west  ?     5  degrees  east  ?     5  degrees  west  ? 

3.  Indianapolis  is  15  degrees  west  of  Boston. 
When  it  is  noon  at  Boston,  what  time  is  it  at  Indian- 
apolis ?  When  it  is  sunset  at  Boston,  where  will  the 
sun  be  at  Indianapolis  ?  , 

4.  Niagara  Falls  are  8  degrees  west  of  Boston. 
When  it  is  noon  at  Boston,  what  time  is  it  at  Niagara 
Falls  ?  When  it  is  4  o'clock  at  Niagara  Falls,  what 
time  is  it  at  Boston  ? 

5.  Washington  city  is  6  degrees  west  of  Boston. 
If  you  set   your  watch  with  the  sun  at  Boston,  and 


LONGITUDE    AND    TIME.  49 

then  carry  it  to  Washington,  your  watch  keeping  ac- 
curate time  all  the  while,  when  you  arrive  at  Wash- 
ington, will  it  be  too  fast  or  too  slow?  and  how 
much  ? 

6.  Two  travellers  met  at  a  public  house.  When 
one  of  them  said  to  the  other,  "  Friend,  are  you  travel- 
ling east  or  west?"  "I  am  direct  from  home,"  said 
the  other,  "  where  niy  watch  agrees  exactly  with  the 
sun,  but  here  I  find  it  is  10  minutes  too  fast.  Now,  if 
you  can  tell  which  way  I  am  travelling,  you  are  wel- 
come to  know." 

Had  he  travelled  east,  or  west  ?  and  how  far  ? 

7.  Boston  is  71  degrees  west  of  London.  When  it 
is  noon  at  Boston,  what  time  is  it  in  London  ? 

8.  The  English  convicts  are  transported  to  Botany 
Bay,  150  degrees  east  of  London.  When  it  is  noon  at 
London,  what  time  is  it  in  Botany  Bay  ? 

9.  English  traders  are  settled  on  Columbia  River, 
120  degrees  west  from  London.  What  time  is  it  there 
when  it  is  noon  in  London  ? 

10.  If  a  man  is  on  the  equator,  which  way  must  he 
travel,  and  how  many  geographical  miles,  to  have  the 
day  4  minutes  longer  than  24  hours  ?  How  far,  to 
have  the  day  2  minutes  longer  ?  How  far,  to  have  it 
1  minute  longer?  How  far  must  he  travel  to  have 
the  day  1  minute  and  a  half  longer?  Which  way 
must  he  travel,  and  how  far,  to  have  the  day  1  minute 
shorter  ?  2  minutes  shorter  ?  5  minutes  shorter  ? 

11.  Suppose  two  birds  start  from  the  same  place  on 
the  equator,  and  fly,  one  east  and  the  other  west,  at 
the  rate  of  60  geographical  miles  an  hour,  and  at  the 
end  of  the  hour  it  is  just  sunset  to  the  bird  flying  east  ; 
how  high  is  the  sun  then  at  the  place  where  the  other 
bird  is  ? 

How  high  was  the  sun  at  the  place  of  their  starting, 
when  they  set  out  ? 

12.  A  shipmaster  sails  from  New  York  for  Europe, 

5  D 


50  MENTAL    ARITHMETIC. 

and  for  three  days  it  is  so  cloudy  that  he  cannot  see 
the  sun.  On  the  fourth  day,  he  takes  an  observation 
of  the  sun  at  noon  j  and  by  his  chronometer,  which 
gives  the  New  York  time,  it  is  half  past  eleven.  How 
many  degrees  east  from  New  York  has  he  sailed  ? 

In  what  longitude  is  he  then,  if  New  York  is  74° 
V  west  from  Greenwich  ? 


SECTION    VII. 

PRIME   NUMBERS. 

Numbers  may  be  divided  into  two  great  classes. 
The  first  class  comprises  such  numbers  as  cannot  be 
formed  by  the  multiplication  of  any  two  or  more  num- 
bers together  ;  as,  1,  2,  3,  5,  11,  17.  These  are  called 
prime  numbers.  The  other  class  may  be  formed  by 
multiplying  two  or  more  numbers  together,  as  4,  which 
is  formed  by  multiplying  2  by  2  ;  6,  which  is  equal  to 
2X3;  10,  which  is  equal  to  2X5,  &c.  These  are 
called  composite  numbers.  These  may  always  be 
formed  by  multiplying  two  or  more  prime  numbers 
together.  Thus  all  numbers  are  either  prime,  or  are 
formed  by  the  multiplication  of  prime  numbers  to- 
gether. 

In  separating  numbers  into  their  factors,  care  should 
be  taken  that  the  factors  be  all  prime.  Thus,  in  re- 
solving 30  into  its  factors,  we  may  say  it  is  formed  by 
multiplying  5  by  6 ;  but  this  is  not  sufficient,  for  6  is 
not  prime  ;  it  is  formed  of  the  factors  2  and  3.  The 
prime  factors  of  30,  therefore,  are  2,  3,  and  5.  We 
may  say  that  30  is  formed  of  the  factors  3  and  10;  but 
here,  again,  the  analysis  is  not  complete,  for  10  is  not 
prime  ;  it  is  composed  of  the  factors  2  and  5.     Thus 


PRIME    AND    COMPOSITE    NUMBERS. 


51 


we  are  brought  to  the  same  three  factors  as  before, 
namely,  2,  3,  and  5. 

The  following  table  of  numbers  from  1  to  100  will 
show  what  of  them  are  prime,  and  what  are  the  prime 
factors  of  those  which  are  composite.  This  table 
should  be  carefully  studied  and  made  perfectly  familiar. 
The  analysis  of  composite  numbers  into  their  prime 
factors  lies  at  the  foundation  of  some  of  the  most  im- 
portant operations  in  numbers,  and  affords  an  insight 
into  some  of  the  most  intricate  rules  of  arithmetic. 


1,  prime. 

27  =  3X3X3. 

2,  prime. 

28  =  2X2X7. 

3,  prime. 

29,  prime. 

4  =  2X2. 

30  =  2X3X5. 

5,  prime. 

31,  prime. 

6  =  3X2. 

32  =  2X2X2X2X2. 

7,  prime. 

33  =  3X11. 

8  =  2X2X2. 

34  =  2X17. 

9  =  3X3. 

35  =  5X7. 

10  =  2X5. 

36  =  2X2X3X3. 

11,  prime. 

37,  prime. 

12  =  2X2X3. 

38  =  2X19. 

13,  prime. 

39  =  3X13. 

14  =  2X7. 

40  =  2X2X2X5. 

15  =  3X5. 

41,  prime. 

16  =  2X2X2X2. 

42  =  2X3X7. 

17,  prime. 

43,  prime. 

18  =  2X3X3. 

44  =  2X2X11. 

19,  prime. 

45  =  5X3X3. 

20  =  2X2X5. 

46  =  2X23. 

21  =  3X7. 

47,  prime. 

22  =  2X11. 

48  =  2X2X2X2X3. 

23,  prime. 

49  =  7X7. 

24  =  2X2X2X3. 

50  =  2x5X5. 

25  =  5X5. 

51  =  3x17- 

y      26  =  2X13. 

52  =  2x2X13. 

52 


MENTAL    ARITHMETIC. 


53,  prime. 

54  =  2X3X3X3. 

55  =  5X11. 

56  =  2X2X2X7. 

57  =  3X19. 
58=2X29. 
59,  prime. 

60  =  2X2X3X5. 
61,  prime. 

62  =  2X31. 

63  =  3X3X7. 

64  =  2X2X2X2X2X2. 

65  =  5X13. 

66  =  2X3X11. 
67.  prime. 

68  =  2X2X17. 

69  =  3X23. 

70  =  2X5X7. 
71,  prime. 

72  =  2X2X2X3X3. 
73,  prime. 

74  =  2X37. 

75  =  3X5X5. 

76  =  2X2X19. 


77  =  7X11. 

78  =  2X3X13. 
79,  prime. 

80  =  2X2X2X2X5. 

81  =  3X3X3X3. 

82  =  2X41. 
83,  prime. 

84  =  2X2X3X7. 

85  =  5X17. 

86  =  2X43. 

87  =  3X29. 

88  =  2X2X2X11. 
89,  prime. 

90  =  2X3X3X5. 

91  =  7X13. 

92  =  2X2X23. 

93  =  3X31. 

94  =  2X47. 

95  =  5X19. 

96  =  2X2X2X2X2X3. 
97,  prime. 

98  =  2X7X7. 

99  =  3X3X11. 
100  =  2X2X5X5. 


On  examining  this  table,  several  things  may  be  ob- 
served. 

n     1.  All  the  even  numbers  are  composite;   for  they 
•are  all  divisible  by  2.     So  it  appears  in  the  table,  with 
the  exception  of  the  number  2,  which  is  regarded  as 
prime  because  it  is  divisible  only  by  itself. 

2.  Several  of  the  numbers  given  above  are  powers 
of  their  prime  factors.  Thus  4  is  the  2d  power  of  2, 
8  the  third  power  of  2,  16  the  4th  power  of  2,  32  the 
5th,  64  the  6th.  9,  27,  and  81,  are  the  2d,  3d,  and 
4th  powers  of  3.  25  is  the  2d  power  of  5,  49  the  2d 
power  of  7. 


PROPERTIES    OF    PRIME    AND    COMPOSITE    NUMBERS.   53 

3.  If  you  double  the  number  of  times  a  factor  is 
taken,  you  obtain  the  square  of  the  number  they  at 
first  made.  Thus  4  is  obtained  by  taking  2  twice  as 
a  factor.  If  you  take  it  twice  as  many  times,  that  is, 
4  times,  as  a  factor,  you  obtain  16,  which  is  the  square 
of  4. 

9  is  obtained  by  taking  3  twice  as  a  factor.  If  you 
double  the  number  of  times  it  is  taken,  thus,  3X3X3 
X3,  you  obtain  the  square  of  9. 

8  is  obtained  by  taking  2  three  times  as  a  factor. 
If  you  take  it  6  times,  you  obtain  64,  the  square  of  8. 

So  universally,  if  you  double  the  number  of  times 
a  factor  is  taken  to  produce  a  certain  number,  you  ob- 
tain, not  twice  that  number,  but  the  square  of  it. 

I  will  make  a  single  remark  here  about  the  prime 
numbers,  and  then  call  your  attention  to  the  compos- 
ite numbers. 

Since  the  prime  numbers  are  not  formed  by  multi- 
plying any  two  or  more  numbers  together,  they  can- 
not be  divided  by  any  number.  You  will  observe, 
however,  that  any  number  whatever  may  be  divided 
by  itself,  and  may  also  be  divided  by  1 ;  but  1  is  a 
unit,  and  not  a  number ;  and  by  dividing  a  number  by 
itself,  or  by  1,  you  obtain  no  new  number.  Dividing 
the  number  by  itself,  you  obtain  1 ;  and  dividing  by  1, 
you  obtain  the  number  itself.  Such  an  operation, 
therefore,  brings  out  nothing  new.  It  is  only  another 
way  of  expressing  what  was  just  as  plain  before.  In 
the  same  way,  we  may  sometimes  regard  a  number 
as  produced  by  multiplying  itself  into  1 ;  thus,  7  =  7 
XI;  but  this  is  not  multiplication,  but  only  an  ex- 
pression in  the  form  of  multiplication.  It  produces 
no  new  number,  and  is  employed  only  for  convenience, 
in  order  to  make  the  reasoning  more  plain. 

Composite  numbers  can  be  divided  by  their  factors. 
Thus  you  can  divide  10  either  by  2  or  by  5,  and  by 
no  other  number.  If  you  divide  by  2,  you  obtain  5 
5* 


54  MENTAL    ARITHMETIC. 

for  the  answer,  or  quotient ;  if  you  divide  by  5,  you 
obtain  2  for  the  answer.  Dividing  by  a  number,  then, 
is  the  same  as  erasing  that  number  as  a  factor,  and 
will  always  give  for  the  answer  the  other  factor,  or 
factors.  Thus,  dividing  10  by  2  you  may  represent 
thus,  $X5,  leaving  the  factor  5  for  the  answer;  di- 
viding 10  by  5,  thus,  2X0,  leaving  2  for  the  answer. 
Divide  21  by  3,  thus,  £X7.  Divide  12  by  3,  thus, 
4X3,  or  $X^X3. 

It  is  plain,  therefore,  that  if  you  express  any  num- 
ber by  its  factors,  you  can\at  once  see  what  numbers 
you  can  divide  it  by.  You  can  divide  it  by  each  of 
its  prime  factors,  or  by  any  combination  of  them,  and 
by  no  other  number.  Thus  6  =  2X3  you  can  divide 
by  2  or  by  3;  8  =  2X2X2  you  can  divide  by  2,  and 
that  quotient  by  2,  and  that  by  2  again;  30  =  2X3 
X5  you  can  divide  by  2,  or  3,  or  5,  or  by  any  two 
of  them  combined. 

Any  composite  number  may  be  divided  by  any  of 
its  prime  factors,  or  by  any  combination  of  them. 

By  what  numbers  can  you  divide  15?  18?  20?  21? 
26?  27?  36?  42?  46?  48?  49?  50? 

Sometimes  we  have  two  numbers,  and  we  wish  to 
know  if  there  is  any  number  that  will  divide  them 
both.  This  we  can  ascertain  if  we  express  each  num- 
ber by  means  of  its  prime  factors,  and  then  see  if  the 
same  factor  is  found  in  both.  If  so,  they  are  both  di- 
visible by  that  number.  Thus,  if  we  wish  to  know 
whether  any  number  will  divide  both  9  and  15,  we 
express  them  thus,  3X3  and  3X5.  Now,  3  appears 
as  a  factor  in  both;  they  can  both,  therefore,  be  di- 
vided by  3.  This  number,  3,  is  called  the  common 
divisor,  because  it  is  a  divisor  common  to  several  num- 
bers. If  we  wish  to  know  whether  any  number  will 
divide  both  15  and  8,  we  express  15  by  its  factors, 
5X3;  and  8  by  its  factors,  2X2X2.  Now,  there  is 
no  factor  common  to  both  ;  no  number,  therefore,  will 


PROPERTIES    OF    PRIME    AND    COMPOSITE    NUMBERS.   55 

divide  them  both ;  in  other  words,  they  have  no  com- 
mon divisor.  Numbers  which  have  no  common  di- 
visor are  said  to  be  prime  to  each  other.  They  may 
be  composite  considered  by  themselves,  as  is  the  case 
with  8  and  15 ;  but  if  they  have  no  common  divisor, 
they  are  said  to  be  prime  to  each  other.  Numbers 
which  have  a  common  divisor  are  said  to  be  compos- 
ite to  each  other.  If  there  are  more  than  two  num- 
bers, they  must  be  treated  in  the  same  way.  Each 
must  be  written  in  the  form  of  its  prime  factors;  and 
then,  if  any  one  number  appears  as  a  factor  in  them 
all,  they  are  divisible  by  that. 

Is  there  any  common  divisor  to  9,  14,  and  27? 
Written  in  the  form  of  their  factors,  they  stand  thus, 
2X3;  2X7;  3X3X3.  They  have,  therefore,  no 
common  divisor;  for,  though  3  or  9  will  divide  both 
the  first  and  the  third  number,  it  will  not  divide  the 
second ;  and  neither  2  nor  7,  which  are  the  factors  of 
the  second  number,  appear  in  the  first  or  third.  9,  14, 
and  27,  are,  therefore,  prime  to  each  other. 

What  is  the  common  divisor  of  15  and  27?  Of  14 
and  22?    Of  21  and  49?    Of  35  and  28?    Of  6  and  21? 

Let  us  now  take  the  following  question :  What  is 
the  common  divisor  of  18  and  30?  By  inspecting 
their  factors,  2X3X3,  and  2X3X5,  we  find  that  2X3, 
or  6,  is  common  to  both ;  6  is  therefore  the  greatest 
common  divisor. 

What  is  the  greatest  common  divisor  of  18  and  27  ? 
Of  4,  8,  and  36?  Of  15  and  45?  Of  27  and  45? 
Of  40,  64,  and  16?  Of  44  and  24?  Of  75  and  15? 
Of  80  and  100?  Of  60  and  24?  Of  35,  21,  and  49? 
Of  15  and  50? 

We  have  seen  that  a  composite  number  can  be  di- 
vided only  by  its  factors,  and  that  prime  numbers  can- 
not be  divided  at  all.     It  is  frequently  necessary,  how- 
ever, to  attempt  the  division  of  prime  numbers,  and  to-, 
divide  composite  numbers  by  some  number  different 


56  MENTAL    ARITHMETIC. 

from  their  factors.  For  example,  we  may  wish  to 
divide  9  by  4,  or  to  obtain  one  fourth  of  9.  Now,  4 
is  not  a  factor  of  9,  and  the  actual  division  of  9  by  4 
is,  strictly  speaking,  impossible.  We  proceed  in  this 
way.  We  divide  8  by  4,  and  obtain  2  for  the  answer, 
and  we  have  a  remainder  of  1,  which  we  have  not 
divided.  To  show  that  we  design  this  to  be  divided 
by  4,  we  write  the  4  under  it,  with  a  line  between, 
thus,  i.  In  this  way  we  indicate  plainly  enough  what 
the  answer  is,  although  we  have  no  one  figure  that 
will  express  it. 

Again,  let  us  divide  15  by  4.  We  divide  12  by  4, 
obtaining  for  the  quotient  3,  and  we  have  3  remaining. 
This  we  cannot  divide  by  4  so  as  to  express  the  quo- 
tient by  a  single  figure.  We  therefore  indicate  the 
division,  without  performing  it,  by  writing  the  divisor, 
4,  under  the  dividend,  3,  thus,  f.  When  we  divide  a 
number  by  4,  we  obtain  one  fourth  of  it.  The  ex- 
pression |,  therefore,  signifies  one  fourth  of  three. 
But  one  fourth  of  three  is  the  same  quantity  as  three 
fourths  of  one. 


Thus 1 1 the    perpendicular   stroke 

cuts  oif, . . on    the    left,    one    fourth 

part    of 1 -, the    three    whole    lines. 

By  looking  at  what  is  thus  cut  off,  you  see  that  it  is 
just  equal  to  the  three  fourths  of  one  whole  line,  on 
the  right  hand  of  the  perpendicular  mark.  We  may, 
therefore,  call  £  one  fourth  of  three,  or  three  fourths 
of  one;  or,  simply,  three  fourths,  meaning  threa 
fourths  of  one;  and,  as  this  is  the  shorter  expres- 
sion, it  is  the  one  usually  employed. 


Questions. 

1.  What  will  be  the  remainder  in  dividing  18  by  5? 
How  will  you  express  this  part  of  the  division,  which     ^ 


FRACTIONS.  57 

it  is  impossible  to  perform?    In  what  different  ways 
can  yon  read  the  expression? 

2.  How  will  you  draw  lines  on  the  board,  or  slate, 
so  as  to  show  that  one  fifth  of  three  is  equal  to  three 
fifths  of  one? 

3.  What  will  be  the  remainder  in  dividing  23  by  6  ? 
How  will  you  express  the  part  of  the  division  which 
is  not  performed?  In  what  different  ways  can  you 
read  the  expression? 

4.  How  will  you  draw  lines,  so  as  to  show  that  one 
sixth  of  five  is  equal  to  five  sixths  of  one  ? 

5.  In  all  the  above  cases,  which  is  the  larger,  the 
dividend,  or  the  divisor? 

6.  In  all  the  above  expressions,  what  is  the  value  of 
the  quantity  expressed,  compared  to  one  ?  Is  it  greater, 
or  less,  than  one  ? 

In  the  same  way  as  in  the  above  cases,  we  may  de- 
note the  division  of  any  number  by  a  number  greater 
than  itself;  by  writing  the  divisor  under  the  dividend; 
as)  fj  fj  t£-  These  expressions  are  called  Fractions. 
A  fraction,  therefore,  is  an  expression  denoting  the 
division  of  a  number  by  a  number  greater  than  itself; 
and  as  the  quantity  signified  by  such  an  expression  is 
necessarily  less  than  one,  we  may  say,  more  briefly, 

A  fraction  is  an  expression  for  a  quantity  less  than 
one. 

It  is  expressed  by  means  of  two  numbers,  —  the 
smaller  (the  dividend)  written  above,  the  larger  (the 
divisor)  written  below,  a  horizontal  line. 

The  number  below  the  line  is  called  the  denom- 
inator, because  it  shows  into  how  many  equal  parts 
the  number,  or  unit,  is  divided ;  the  number  above  the 
line  is  called  the  numerator,  because  it  shows  how 
many  units  are  divided,  or  how  many  parts  of  a  di- 
vided unit  are  taken. 


58  MENTAL.    ARITHMETIC. 

7.  In  the  expressions  f ,  f,  -fT,  is  the  quantity  denoted 
in  each  expression  greater  than  1,  or  less?  and  why? 

8.  In  the  expressions  f ,  £,  f,  is  the  quantity  denoted 
in  each  case  greater  than  1,  or  less?  and  why? 

It  is  sometimes  convenient  to  express  quantities 
which  are  not  less  than  1,  in  the  form  of  fractions; 
as,  f,  f,  &c.  These  are  called  Improper  Fractions^ 
while  the  others  are  called,  in  distinction  from  these, 
Proper  Fractions.  An  improper  fraction  may  always 
be  reduced  to  a  whole  number  and  a  proper  fraction. 

A  whole  number  and  a  fraction  taken  together,  is 
called  a  Mixed  Number. 

9.  Reduce  to  a  mixed  number,  -V3-;  jtS'  ft- 

10.  Reduce  to  an  improper  fraction,  4|j  7£;  9|. 

Questions. 

What  is  meant  by  a  common  divisor? 

What  is  meant  by  the  greatest  common  divisor? 

When  are  numbers  prime  to  each  other? 

When  are  numbers  composite  to  each  other? 

What  is  the  process  of  dividing  13  by  4? 

In  dividing  16  by  5?     In  dividing  25  by  6? 

What  are  Fractions  ? 

Explain  what  is  signified  by  each  of  the  numbers 
in  the  fraction  §.     In  \.    In  T6T.    In  tV*    In  f .    In  T£. 

A  man  bought  a  barrel  of  flour,  and  gave  away  two 
fifths  of  it.  What  fraction  will  express  what  he  gave 
away?     What  fraction  will  express  what  he  kept? 

A  man  bought  a  load  of  hay,  and  sold  two  elevenths 
of  it.  What  fraction  will  express  what  he  sold? 
What  fraction  will  express  what  he  kept? 

What  is  a  proper  fraction  ?    Give  an  example. 

What  is  an  improper  fraction?    Give  an  example. 

When  is  the  value  of  a  fraction  just  equal  to  1  ?  < 


FRACTIONS.  59 

SECTION    VIII. 

MULTIPLICATION   AND    DIVISION   OF   FRACTIONS. 

We  have  seen  that  a  fraction  is  not  a  simple  ex- 
pression, but  composed  of  two  numbers;  and  its  value 
cannot  be  determined  by  one  of  these  numbers  alone, 
but  by  both  taken  in  connection.  By  looking  at  the 
numerator,  you  cannot  tell  the  value  of  the  fraction, 
unless  you  know  what  the  denominator  is.  By  look- 
ing at  the  denominator,  you  cannot  tell  the  value  of 
the  fraction,  unless  you  know  what  the  numerator  is. 

Let  us  now  observe  the  effect  of  altering  one  of  the 
terms  of  the  fraction  without  altering  the  other.  We 
will  take  the  fraction  f.  If  we  increase  the  numer- 
ator by  1,  making  it  f,  we  increase  the  value  of  the 
fraction,  for  we  take  one  fifth  more  than  we  had  be- 
fore. So,  if  we  multiply  the  numerator  by  2,  making 
it  |,  we  double  the  value  of  the  fraction ;  and  so  of 
any  other  numbers,  if  we  multiply  the  numerator,  we 
multiply  the  value  of  the  fraction.  And,  by  the  same 
reasoning,  if  we  divide  the  numerator  by  2,  we  divide 
the  fraction  by  2,  for  |  is  plainly  one  half  as  great  as 
f.  So  of  all  other  numbers,  by  dividing  the  numer- 
ator we  divide  the  fraction. 

Let  us  now  observe  the  effect  of  altering  the  denom- 
inator. If  we  increase  the  denominator  of  the  fraction  § 
by  1,  making  it  f,  we  have  not  increased  the  fraction, 
but  diminished  it;  for  one  sixth  is  less  than  one  fifth, 
and  any  number  of  sixths  are  less  than  the  same  num- 
ber of  fifths.  We  will  multiply  the  denominator  of  the 
fraction  f  by  2,  making  T2n.  What  effect  has  been 
produced  on  the  value  of  the  fraction?  One  tenth 
is  half  as  great  as  one  fifth,  and  two  tenths  are  half  as 
great  as  two  fifths.     The  fraction  is,  .therefore,  half 


00  MENTAL    ARITHMETIC. 

as  great  as  it  was  before ;  that  is,  it  has  been  divided 
by  2.  Multiplying  the  denominator,  therefore,  divides 
the  value  of  the  fraction. 

We  will  now  divide  the  denominator.  Take  the 
fraction  f ;  dividing  the  denominator  by  2,  we  have  f- . 
Now,  as  this  is  twice  as  great  as  f ,  we  have  multiplied 
the  fraction,  by  dividing  the  denominator. 

There  are,  then,  two  ways  of  multiplying  a  fraction. 
We  may  multiply  the  numerator;  or,  if  the  multiplier 
is  a  factor  of  the  denominator,  we  may  divide  the  de- 
nominator. Thus,  to  multiply  f  by  2,  we  may  multi- 
ply the  numerator,  which  gives  f ,  or  divide  the  denom- 
inator, which  gives  f ,  equal  to  f . 

To  divide  a  fraction,  we  may  either  divide  the 
numerator,  if  the  divisor  is  a  factor  of  it,  or  we  may 
multiply  the  denominator.  Thus,  to  divide  f  by  3, 
we  may  divide  the  numerator,  giving  f ,  or  we  may 
multiply  the  denominator,  which  gives  /T,  which  is 
equal  to  f . 

We  will  now  multiply  both  terms  of  the  fraction  by 
the  same  number.  Multiplying  both  terms  of  the 
fraction  §  by  3,  we  have  f.  Here  the  denominator, 
expressing  the  number  of  parts  into  which  the  unit  is 
divided,  is  three  times  as  great  as  it  was  before ;  con- 
sequently each  of  the  parts  is  only  one  third  as  great ; 
but  the  numerator  has  also  been  multiplied  by  3,  so 
that  three  times  as  many  parts  are  taken,  and  this 
makes  the  value  of  the  fraction  just  equal  to  what  it 
was  before.  So  we  may  multiply,  by  any  number 
whatever,  both  terms  of  the  fraction  f,  and  the  value 
will  still  be  the  same  as  before ;  for  example,  £,  f ,  T8j, 
i£>  If  j  eacn  °f  which  is  equal  to  §.  We  may,  then,  at 
any  time  multiply  both  terms  of  a  fraction  by  the  same 
number,  without  altering  the  value  of  the  fraction. 
By  the  same  reasoning,  we  may  divide  both  terms  of 
a  fraction  by  the  same  number  without  altering  its 
value.     Taking  the  examples  above,  we  may  divide    ^ 


FRACTIONS.  61 

the  terms  of  f  by  2,  and  we  obtain  f ;  dividing  the 
terms  of  f  by  3  gives  us  § ,  and  so  of  the  others,  §  is 
the  same  fraction  as  $,  f,  t8?,  &c,  but  it  is  expressed  in 
lower  terms,  and  therefore  is  more  convenient.  It  is 
easier  to  write  ^  than  it  is  to  write  Jf,  though  both 
have  the  same  value. 

To  reduce  a  fraction  to  its  lowest  terms,  we  divide 
both  the  numerator  and  denominator  by  their  greatest 
common  divisor.  To  find  the  greatest  common  divi- 
sor, separate  each  term  into  its  prime  factors,  and  erase 
those  which  are  common  to  both.  The  remaining 
factors  will  express  the  value  of  the  fraction  in  its 
lowest  terms. 

Treating  the  above  fractions  in  this  way,  they  ap- 
pear thus, 
4_2X2  6_2X2  8  _2X^X2  10__2X#  12_gX2X# 

6~^X3'  9~  3X?  12~2X#X3'  To~  3X?  18~~ ft  X  3  X  0' 
leaving,  in  each  case,  f. 

In  how  many  ways  can  you  obtain  the  answer  to  the 
following  questions  ?     |X2?     fX3?     fx4?     t^X2? 

In  how  many  ways  can  you  obtain  the  answer  to  the 
following?  fX3?  |X2?  fX4?  t2tX6?  tVX5? 
AX3? 

In  how  many  ways  can  you  obtain  the  answer  to 
the   following?      f-i-3?      f-r-4?      -ft-r-3?.    if-^-2? 

In  how  many  ways  can  you  obtain  an  answer  to 
the  following?  f-r-2?  f -=-4?  i-^3?  t9<j-^2? 
t§-^5? 

Reduce  to  their  lowest  terms  each  of  the  following 
fractions :  f;  J+;  *;  §§;  f$;  U]  U]  ¥t)  ft 5  A ) 
is  i  M  J  s8^  )  If  J  it • 

TO   FIND  THE    DIVISORS   OF  NUMBERS. 

Reduce  the  fraction  {£§  to  its  lowest  terms. 
You  will  not  see  immediately  that  these  two  num- 
6 


b2  MENTAL    ARITHMETIC. 

bers  have  any  common  divisor.  To  assist  yon  to 
reduce  fractions  of  this  kind,  something  will  here  be 
said  about  the  way  of  finding  the  divisors  of  num- 
bers. Let  us  first  inquire  what  numbers  can  be  divi- 
ded by  2. 

We  have  seen  that  all  even  numbers,  and  only  those, 
can  be  divided  by  2. 

What  numbers  can  be  divided  by  4  ? 

If  you  examine,  you  will  find  that  all  even  tens  are 
divisible  by  4 ;  as,  20,  40,  60,  &c.  If,  therefore,  the 
tens  are  even,  and  the  units  are  divisible  by  4,  then  the 
whole  is  divisible  by  4.  But  the  only  unit  numbers 
divisible  by  4  are  4  and  8 ;  therefore,  if  the  tens  are 
even,  and  the  unit  number  is  4  or  8,  the  whole  is  di- 
visible by  4;  as,  84,  88;   124,  128;   148,  364;  &c. 

Again;  as  10,  when  divided  by  4,  leaves  a  remainder 
of  2,  any  odd  number  of  tens  will  do  the  same  ;  as,  30, 
50,  70,  90;  for  every  odd  number  of  tens  is  an  even 
number  of  tens +  10.  If,  then,  the  number  of  tens  is 
odd,  the  units  must  be  two  less  than  4  or  8,  in  order 
to  be  divisible  by  4;  that  is,  if  the  tens  are  odd,  and 
the  units  2  or  6,  the  whole  is  divisible  by  4 ;  as,  72, 
96,  52,  &c. 

Are  the  following  even  numbers  divisible  by  4,  or 
only  by  2?  and  why?  126;  82;  94;  92;  138;  156; 
346;  548;  76;  58;  392. 

What  numbers  can  be  divided  by  8? 

As  100  divided  by  8  leaves  a  remainder  of  4,  (8X12 
=  96,)  it  follows  that  200  will  be  exactly  divisible  by 
8,  for  the  two  remainders  of  4  will  make  8.  If  200 
is  divisible  by  8,  it  follows  that  all  even  hundreds  are 
divisible  by  8  ;  as,  400,  600,  1400,  &c. 

If,  therefore,  the  hundreds  are  even,  and  the  tens 
and  units  are  divisible  by  8,  the  whole  number  will  be 
divisible  by  8;  as,  248,  672,  1456,  &c. 

Again ;  if  the  hundreds  are  odd,  and  the  tens  and 
units  are  4  less  than  some  multiple  of  8,  the  whole 


FRACTIONS.  63 

number  will  be  divisible  by  8  j  for  the  odd  hundred, 
divided  by  8,  leaves  a  remainder  of  4 ;  and  this,  added 
to  the  tens  and  units,  will  make  an  exact  multiple  of  8. 

Are  the  following  numbers  divisible  by  8,  or  by  4  ? 
and  why?  444;  944;  136;  1328;  712;  532;  816; 
516;  384;   128;   1236. 

What  numbers  are  divisible  by  5  ?  All  tens  are  di- 
visible by  5  ;  consequently,  if  the  unit  figure  is  5  or  0, 
the  whole  number  is  divisible  by  5. 

What  numbers  are  divisible  by  3  ?  By  examining 
the  multiples  of  3,  we  shall  find  this  singular  fact,  that 
the  sum  of  the  figures  which  express  any  multiple  of  3 
is  itself  a  multiple  of  3.  Take  the  multiples  of  3  from 
12  to  24;  12,  15,  18,  21,  24;  by  adding  the  figures 
which  express  any  one  of  these  multiples,  we  find  that 
the  sum  is  a  multiple  of  3.  The  figures  of  12  added 
are  1+2  =  3,  of  15  are  1  +  5  =  6,  of  18  are  1  +  8  = 
9,  of  21  are  2+ 1  =  3,  of  24  are  2  +  4  =  6.  The  same 
is  true  of  all  multiples  of  3. 

It  will  also  be  found,  that,  if  you  add  the  figures  of 
any  number,  and  the  sum  is  a  multiple  of  3,  the  whole 
number  is  a  multiple  of  3.  To  know,  then,  if  a  num- 
ber is  a  multiple  of  3,  add  together  the  figures  that 
express  the  number,  and  if  the  sum  is  a  multiple  of 
3,  the  whole  number  is  a  multiple  of  3. 

Are  the  following  numbers  divisible  by  3  ?  471 ;  59  ; 
115;  642;  624;   138;  234;  742;  894. 

It  follows  from  what  has  been  said,  that,  if  any 
number  is  divisible  by  3,  any  other  number  expressed 
*  by  the  same  figures  differently  arranged  will  also  be 
divisible  by  3 ;  for  the  sum  made  by  adding  the  fig- 
ures will  be  the  same  in  whatever  order  they  are 
taken. 

Thus,  if  936  is  divisible  by  3 ;  369,  396,  963,  639, 
1  693,  are  each  divisible  by  3. 

We  will  next  inquire  what  numbers  are  divisible 
»  by  6.     As  6  =  2X3,  any  number  that  is  divisible  by 


64  MENTAL    ARITHMETIC. 

2  and  by  3  is  divisible  by  6.  You  have  learned  what 
numbers  are  divisible  by '3,  and  what  by  2.  If  a 
number  combines  both  these  conditions,  it  is  divisible 
by  6 ;  that  is,  all  numbers  are  divisible  by  6,  the  sum 
of  whose  figures  is  a  multiple  of  3,  and  whose  last 
figure  is  an  even  number. 

What  combinations  of  the  figures  1,  2,  3,  will  give 
numbers  divisible  by  6  ?    and  what  by  3  only  ? 

Next  let  us  inquire  what  numbers  are  divisible  by  9. 

If  the  figures  which  express  any  multiple  of  9,  as 
18,  27,  36,  45,  54,  be  added  together,  the  sum  will  be 
a  multiple  of  9. 

Also,  if  the  figures  of  any  number  be  added  to- 
gether, and  the  sum  is  a  multiple  of  9,  the  whole 
number  is  divisible  by  9. 

Are  the  following  numbers  divisible  by  9  ?  and 
why?  936;  972;  396;  423;  387;  527;  441;  416; 
315;  756. 

Any  number  divisible  by  9  and  by  2  is  divisible  by 
9X2,  or  18.  Which  of  the  above  numbers  are  divis- 
ible by  18  ? 

Any  number  divisible  by  9  and  by  4  is  divisible  by 
9X4,  or  36.  Which  of  the  above  numbers  are  divis- 
ible by  36  ?  ^ 

Any  number  divisible  by  9  and  by  8  is  divisible  by 
9X8,  or  72.  Is  either  of  the  above  numbers  divisible 
by  72? 

Any  number  divisible  by  9  and  by  5  is  divisible  by 
9X5,  or  45.  Which  of  the  above  numbers  is  divisi- 
ble by  45  ? 

What  are  divisors  of  124?  Of  176?  Of  252?  Of 
384?  Of  153?  Of  186?  Of  207  ?  Of  702?  Of 
4041? 

We  will  now  return  to  the  fraction  that  was  first 
given.     Reduce  f|f  to  its  lowest  terms. 

Reduce  to  lowest  terms,  T££;  g|f ;  J$f. 

Reduce  to  lowest  terms,  fff ;  %{$;  Iff. 


FRACTIONS.  65 

SECTION    IX. 

MULTIPLICATION   OF   FRACTIONS   BY   FRACTIONS. 

We  have  seen  how  we  may  multiply  or  divide  a 
fraction  by  a  whole  number.  We  will  now  inquire 
how  we  can  multiply  or  divide  one  fraction  by  another. 
Let  us  multiply  $  by  f.  First  multiply  f  by  2,  wrhich 
gives  f-  for  the -answer.  But  here  we  have  multiplied 
by  2,  instead  of  the  real  multiplier,  f.  Now,  2  is  5 
times  greater  than  f;  the  product  f,  then,  is  5  times 
greater  than  it  should  be.  It  must  therefore  be  di- 
vided by  5.  We  divide  f  by  5  by  multiplying  the 
denominator  by  5,  giving  ^  for  the  answer. 

In  the  same  way  multiply  -|  by  T\;    f  X|;    |-X£. 

DIVISION   OF   FRACTIONS   BY   FRACTIONS. 

Let  us  now  divide  f  by  f.  First  divide  f  by  3. 
This  we  do  by  multiplying  the  denominator  by  3, 
giving  for  the  answer  TV  Here,  however,  we  have 
divided  by  3,  instead  of  the  true  divisor,  f .  We  have 
used  a  divisor  seven  times  too  large.  The  quotient, 
therefore,  will  be  seven  times  too  small;  T\  must 
therefore  be  multiplied  by  7,  making  the  answer  ff. 
In  the  same  way  perform  the  following:  §  —  £;  f-~$ ; 
?-H-;  f-H;  i-H. 

The  above  analysis  shows  the  grounds  of  the  rules 
usually  given  in  arithmetics  for  the  multiplication  and 
division  of  fractions. 

For  Multiplication,  multiply  the  numerators  to- 
gether for  a  new  numerator,  and  the  denominators 
for  a  new  denominator. 

For  Division,  invert  the  divisor,  and  proceed  as 
in  multiplication. 

Sometimes  we  wish  to  find  the  value  of  a  eom- 
6  *  E 


66  MENTAL    ARITHMETIC. 

pound  fraction,  as  §  of  f .  In  such  cases,  we  may  un- 
derstand the  sign  of  multiplication,  X,  to  stand  in  the 
place  of  the  word  of,  and  treat  it  as  a  case  of  multi- 
plication; for,  in  the  above  example,  it  is  plain  that 
one  third  off  is  T%]  and  two  thirds  is  twice  as  much, 
that  is,  T%. 

What  is  f  of  |  of  f  ?  Multiplying  as. we  have  done 
above,  we  have  for  the  answer  t2<jV  But  this  opera- 
tion may  be  shortened.  We  see  that  4  appears  as  a 
factor  both  in  the  numerator  and  the  denominator. 
We  may,  then,  cancel  them  both,  which  will  have  the 
same  effect  as  dividing  both  terms  of  the  answer  by  4. 
Again ;  3  appears  in  both,  the  numerator  and  the  de- 
nominator; for  in  the  denominator  it  is  a  factor  of  9. 
We  may  therefore  cancel  3  in  both  terms. 

2    $     4 
The  question  will  then  appear  thus,  75X-X-5,  sub- 

stituting  3  in  place  of  the  9.  Multiplying  together  the 
terms  that  now  remain,  we  have  §  for  the  answer. 
This  is  the  same  fraction  as  T2^V  If  you  separate  the 
terms  of  T%\  into  their  prime  factors,  and  cancel  what 
are  common  to  both,  the  remaining  factors  will  give 
the  fraction  f . 

Multiply  the  fractions  f  Xf  XTf  Xf  Writing  the 
terms  that  are  composite  in  the  form  of  their  prime  fac- 
tors, and  cancelling  factors  that  are  common  in  both,  it 

#  $       %X7t    $     ^  . '        . 

will  stand  - — - — -x-3 — „X-^ X^,  which  gives  TV 

$X£X2     $X3^3X£     f  & 

Multiply  fXfXf;  ?iX«XJ. 

Multiply  §£XtV:  t|Xz8t;  HXfXf. 


TO  MULTIPLY   OR   DIVIDE   WHOLE   NUMBERS  BY 
FRACTIONS. 

The  above  examples  will  show  how  to  multiply  or 
divide  a  whole  number  by  a  fraction. 


FRACTIONS.  07 

Multiply  7  by  |.  Multiplying  7  by  4  gives  28, 
which  is  5  times  too  great,  because  4  is  five  tijnes 
greater  than  f.  We  must  therefore  divide  the  answer 
by  5,  thus,  -2^.  As  this  is  more  thau  1,  we  can  re- 
duce it  to  a  whole  number  and  a  fraction.  As  f  is 
equal  to  1,  ^  will  be  equal  to  5;  -258-,  therefore,  is  equal 
to  5f. 

In  this  way  multiply  6  by  f- ;  9  by  f ;  8  by  f . 

This  operation  is  in  fact  the  same  as  multiplying  a 
fraction  by  a  whole  number,  which  has  been  treated 
of  already. 

Let  us  next  divide  7  by  f .  Dividing  7  by  3,  we 
have  £.  Here,  however,  we  have  divided  by  a  number 
4  times  too  great;  for  3  is  four  times  greater  than  £. 
If  the  divisor  is  4* times  too  great,  the  quotient  will  be 
4  times  too  small;  f,  therefore,  must  be  multiplied  by 
4,  giving  2u8-  for  the  answer. 

Divide  8  by  f;  9-rfi   ll«Hj   *<B-Aj 

To  reduce  an  improper  fraction,  as  \3-,  to  a  whole 
number  and  a  proper  fraction,  we  have  only  to  con- 
sider how  many  whole  ones  the  fraction  is  equal  to, 
and  how  much  remains.  Thus,  -^  is  equal  to  3; 
-V3-,  therefore,  is  equal  to  3£. 

Reduce  §;  ->/-;  f;  -^-;  tf;  Vj  «J  U]  ¥• 

In  like  manner,  if  we  have  a  whole  number  and  a 
fraction,  we  may  always  reduce  it  to  an  improper 
fraction. 

ADDITION   AND   SUBTRACTION   OF   FRACTIONS. 

Suppose  we  wish  to  add  together  3£,  so  that  its 
value  shall  be  expressed  in  a  single  expression;  we 
must  change  3  to  halves,  which  will  be  f ;  adding  £ 
to  this,  we  have  £  for  the  answer. 

In  order  to  unite  separate  numbers  into  one  expres- 
sion, they  must  be  of  the  same  kind.  We  cannot  unite 
'2  bushels  and  3  pecks  in  one  expression.     It  is  still  2 


G8  MENTAL    ARITHMETIC. 

bushels  and  3  pecks,  and  we  can  make  nothing  else  of 
it.  wBut  if  we  change  the  bushels  to  pecks,  making  8 
pecks,  we  can  then  add  the  3  pecks,  and  bring  it  all  into 
one  expression,  11  pecks.  So,  to  unite  5§,  we  must 
change  the  5  to  thirds,  making  -V5-,  and  add  the  f, 
making  -y%  This  is  called  reducing  a  mixed  number 
to  an  improper  fraction. 

Reduce  to  an  improper  fraction,  7£;  8£;  4f ;  5£; 
6j;  9£;  3§;  5?;   154;   16f;   13$;  20f;  &1£ 

Supposing  we  wish  to  add  £  to  £,  we  must  change 
the  £  to  fourths,  making  f ;  adding  these,  we  have  £ 
for  the  answer.  > 

Add  4  to  TV     J  =  & ;   ft  +  TV  =  A,  -4ws. 

Add|toT3¥;   f+A;  *+&;  *  +  ,&;  $  +  «. 

Let  us  now  add  §  and  f«  This  question,  you  per- 
ceive, has  a  difficulty  which  the  former  ones  had  not; 
for  §  is  no  number  of  fifths,  and  therefore  we  cannot 
bring  the  fraction  into  fifths  by  any  multiplication. 
We  want  a  number  for  the  denominator  which  can  be 
divided  both  by  3  and  by  5.  Now,  if  you  examine, 
you  will  find  no  such  number  until  you  come  to  15. 
This  is,  of  course,  divisible  by  3  and  by  5,  for  these 
are  its  factors.  We  will  then  take  15  for  the  de- 
nominator. This  we  call  the  common  denominator. 
Taking,  now,  the  fractions  §  and  $,  and  changing  the 
denominator,  3,  to  15,  we  see  that  we  have  made  it 
5  times  as  large  as  it  was  before;  that  is,  we  have 
multiplied  it  by  5.  We  must  therefore  multiply  the 
numerator  by  5,  to  preserve  the  value  of  the  fraction. 
The  fraction  §-  then  becomes  T§,  without  altering  its 
value.  Passing,  now,  to  the  second  fraction,  f,  we  see 
that,  in  changing  the  denominator  to  15,  we  have 
multiplied  it  by  3;  we  must  therefore  multiply  its 
numerator  by  3.  This  will  make  the  fraction  Tf. 
The  two  fractions  will  stand,  then,  t£  +  t|,  which 
added  together  are  ??=lrV 


FRACTIONS.  69 


TO  FIND   A   COMMON   DENOMINATOR. 

We  can  always  obtain  a  common  denominator,  by 
multiplying  all  the  denominators  together.  Then, 
for  the  numerators,  consider,  in  the  case  of  each  frac- 
tion, what  its  denominator  has  been  multiplied  by,  in 
order  to  change  it  to  the  common  denominator,  and 
multiply  the  numerator  by  the  same  number.  Thus 
each  fraction  will  have  had  its  numerator  and  its  de- 
nominator multiplied  by  the  same  number,  and  so  its 
value  will  not  be  changed. 

What  is  the  value  of  £  +  *?  Of§  +  j?  Off  +  |? 
Of*+|?    Ofi  +  f?    Off  +  f?    Off  +  f?    Off  +  f? 

Supposing  we  wish  to  add  the  fractions  £  and  £; 
we  can  proceed  as  above,  and,  with  the  common  de- 
nominator, 24,  the  fractions  will  be  Jf+ff.  But  we 
need  not  employ  so  large  a  denominator  as  24.  We 
seek  the  smallest  denominator  that  shall  contain  both 
4  and  6  as  a  factor.  If,  now,  we  separate  4  and  6 
into  their  prime  factors,  we  shall  find  the  factor  2 
belonging  both  to  4  and  to  6;  thus,  2X2,  2X3. 
Now,  one  of  these  may  be  cancelled,  and  we  shall 
still  have  2X2  for  the  number  4,  and  2X3  for  the 
number  6.  Multiplying  the  factors  which  remain, 
2X2X3,  we  have  12  for  the  smallest  common  de- 
nominator. 

From  this  we  see,  that,  when  both  the  denomina- 
tors contain  the  same  factor,  we  may  reject  it  from 
one  of  them,  and  multiply  together  the  factors  that 
remain. 

Add  }  to  T52-.  Here  2X2  is  common  to  both  de- 
nominators. Rejecting  it  in  one,  and  multiplying, 
we  obtain  24  for  the  least  common  denominator. 

Add  T5s  t0  uV  Here  3x3  is  common  to  both  de- 
nominators. Rejecting  it  in  one,  and  multiplying  what 
remains,  we  have  54  for  the  least  common  denominator. 


70  MENTAL    ARITHMETIC. 

Add  JJ  to  &.     Add  §  to  &.     Add  T\  to  &. 

When  more  fractions  than  two  are  to  be  added,  it  is 
often  most  convenient  to  add  two  together  first,  and 
then  add  a  third  to  the  sum  of  these,  and  so  on. 

Add  §  +  F  +  I-  First  add  %  and  £,  which  equal  f. 
Next,   |'+t-    »  =  «;  and  ft=&;    t2  +  A  =  t!  =  1tV, 

Add  £+§+tV  First  add  {  and  & ;  then  to  the 
sum  of  these  add  J. 

Addf+A+t.     Addf  +  f  +  A-     Addf  +  f  +  f. 

Add  A + J.     AddT\  +  f     AddrV  +  ^  +  yV 

From  \±  subtract  T5Z.  From  %  sub.  TV.  From  T£ 
sub.  35j- 

From  Tf  sub.  £.     From  ff  sub.  tV     From  §£  sub.  T9v. 

Miscella neons  Examp les. 

1.  A  man  spends  |  of  a  dollar  in  a  day.  What  part 
of  a  dollar  will  he  spend  in  5  days?  How  much  will 
he  spend  in  9  days?     How  much  in  11  days? 

2.  A  man  earns  f  of  a  dollar  in  a  day.  How  much 
will  he  earn  in  half  a  day  ?  How  much  in  J  of  a  day  ? 
How  much  in  £  of  a  day  ? 

Here  consider  whether  you  can  divide  the  numerator. 

3.  A  man  earns  I  of  a  dollar  in  a  day.  How  much 
can  he  earn  in  half  a  day  ?  How  much  in  £  of  a  day  ? 
How  much  in  l  of  a  day? 

Consider  whether  you  can  divide  the  numerator, 
and,  if  you  cannot,  what  you  must  do. 

4.  A  vessel  filled  with  water  leaks  so  that  f  of  its 
contents  will  leak  out  in  a  week.  At  this  rate,  what 
part  will  leak  cut  in  a  day? 

What  is  }  of  |  ? 

5.  If  a  team  ploughs  £  of  an  acre  in  6  hours,  how 
much  will  it  plough  in  one  hour?  How  much  in  3 
hours  ? 


FKACTIOxNS.  71 

What  is  i  of  f  ?     What  is  £  of  f  ? 

6.  If  a  horse  runs  ^  of  a  mile  in  one  minute,  how 
far  will  he  run  in  f  of  a  minute  ? 

How  far  will  he  run  in  f  of  a  minute  ? 
What  is  £  of  J  ?     What  is  f  of  i  ? 

7.  A  man  has  f-  of  a  dollar,  which  he  wishes  to  dis- 
tribute equally  among  several  persons,  giving  T%  of  a 
dollar  to  each.  How  many  can  receive  this  sum  ?  and 
what  will  be  the  remainder? 

How  many  times  is  TV  contained  in  7  ?  T%  in  7  ?  T\  in  £■ ? 
How  many  times  is  TV  contained  in  4  ?  T23  in  4  ?  T2^  in  f-  ? 
How  many  times  is  5V  contained  in  6  ?  ^  in  6  ?  jfo  in  f-  ? 

8.  A  man  gave  ^  of  a  bushel  of  oats  to  some  horses, 
giving  to  each  |-  of  a  bushel.  To  how  many  did  he 
give  it  ?    and  what  was  the  remainder  ? 

How  many  times  will  TV  go  in  5  ?  In>f  ?  How 
many  times  will  T2F  go  in  f  ? 

9.  A  man  has  £■  of  a  dollar.  He  gives  £  of  a  dol- 
lar to  one  person,  and  f  of  a  dollar  to  a  second.  What 
part  of  a  dollar  has  he  left  ? 

How  many  cents  had  he  at  first  ?  How  many  cents 
did  he  give  away  ?     How  many  cents  had  he  left  ? 

10.  If  13  pounds  of  figs  cost  |  of  a  dollar,  what  is 
that  a  pound? 

11.  If  5£  lbs.  of  figs  cost  T52-  of  a  dollar,  what  is 
that  a  pound?  Find  first  what  one  half  pound  will 
cost. 

12.  If  f  of  a  cwt.  of  iron  cost  4£  dollars,  what  will 
a  hundred  weight,  cost  ? 

13.  If  34^  lbs.  of  tea  cost  llf  dollars,  what  will  1 
pound  cost? 

Here  you  find  -6^9-  pounds  cost  V  °f  a  dollar.  There- 
fore 69  pounds  must  cost  V  of  a  dollar. 

14.  If  §  of  a  barrel  of  Hour  cost  3|  dollars,  what  is 
that  a  barrel  ?  / 

15.  If  wood  is  5}  dollars  a  cord,  what  will  tb  of  & 
cord  cost  ?     What  will  4£  cords  cost  ? 


72  MENTAL    ARITHMETIC. 

16.  If  33£  gals,  of  molasses  cost  llf  dollars,  what 
is  that  a  gallon  ? 

17.  If  31^  gals,  of  vinegar  cost  4f  dollars,  what  is 
that  a  gallon  ? 

18.  If  a  bottle  of  wine,  containing  l£  pints,  cost  | 
of  a  dollar,  what  would  a  barrel  of  wine  come  to  at 
that  rate  ? 

19.  In  a  pile  of  wood  there  are  13^  cords.  How 
many  loads,  of  £  of  a  cord  each,  are  there  in  the  pile? 

20.  How  many  times  will  2£  go  in  7V.  In  9£? 
In  11? 

21.  How  many  loaves,  of  8|  oz.  of  flour  each,  can 
be  made  from  7  pounds  of  flour  ?  « 

22.  If  a  family  consume  3±  pounds  of  flour  a  day, 
how  long  will  a  barrel  of  flour,  that  is,  196  pounds, 
last  them? 

How  long  will  it  last  if  they  consume  2£  lbs.  a  day  ? 

23.  If  a  barrel  of  flour  last  a  family  40  days,  how 
long  will  14  pounds  last  them? 

24.  A  garrison  of  100  men  is  allowed  12  oz.  of  flour 
a  day  to  each  man.  How  long  will  10  barrels  last 
them  ? 

25.  Two  men  hire  a  horse,  a  week,  for  5  dollars. 
One  travels  with  him  30  miles,  the  other  45  miles. 
What  ought  each  to  pay  ? 

26.  Two  men  hire  a  pasture  in  common  for  $4.80. 
One  pastures  his  horse  in  it  1\  weeks ;  the  other  pas- 
tures his  horse  9  weeks.     What  ought  each  to  pay? 

I     27.  A  boy  bought  3  doz.  of  oranges  for  37^-  cents, 
and  sold  them  for  l£  cents  apiece.    What  did  he  gain  ? 

28.  A  man  bought  7  yds.  of  cloth  for  16  dollars,  and 
sold  it  for  3  dollars  a  yard  ?  What  did  he  gain  on 
each  yard  ? 

29.  A  man,  worth  1690  dollars,  left  f  of  his  prop- 
erty to  his  wife.  How  much  did  she  receive  ?  The 
remainder  he  divided  equally  among  3  sons.  What 
did  each  one  receive  ? 


FRACTIONS.  73 

30.  A  man  bequeathed  his  estate  of  14,000  dollars, 
one  third  to  his  wife,  and  the  remainder  to  be  divided 
equally  among  four  sons.  What  did  the  wife,  and  what 
did  each  son,  receive  ? 

31.  In  an  orchard  one  third  of  the  trees  bear  apples, 
two  fifths  as  many  bear  plums,  and  the  rest  bear  cher- 
ries. What  portion  of  the  trees  bear  plums?  What 
portion  bear  cherries  ?  The  number  of  cherry-trees  is 
40.  What  is  the  whole  number  of  trees  in  the  or- 
chard ? 

32.  What  is  f  of  549  ?     What  is  f  of  374  ? 

33.  What  is  i  of  175i  ?     What  is  |  of  198  ? 

34.  What  is  £  of i  of  1640  ?     What  is  f  of  972? 

35.  If  2  barrels  of  flour  cost  ll£  dollars,  what  will 
17  barrels  cost  ?     What  will  22£  barrels  cost  ? 

36.  If  2£  cords  of  wood  cost  15  dollars,  what  will 
68|  cords  cost  ?     What  will  200  cords  ? 

37.  If  a  horse  eat  2£  tons  of  hay  in  30  weeks, 
what  part  of  a  ton  will  he  eat  in  1  week  ? 

38.  What  is  the  cost  of  23£  yds.  of  cloth  at  £  of  a 
dollar  a  yard  ? 

39.  What  is  the  cost  of  31|  gallons  of  molasses,  at 
T%  of  a  dollar  a  gallon  ? 

40.  A  grocer  drew  from  a  cask,  containing  31^  gal- 
lons, i  of  its  contents.  Now,  how  much  did  he  draw 
out  ?     How  much  remained  ? 


SECTION    X. 

THJE  LEAST   COMMON   MULTIPLE. 

Name  the  multiples  of  2  up  to  20. 
Name  the  multiples  of  5  up  to  30. 
Name  the  multiples  of  7  up  to  63. 
Name  the  multiples  of  3  up  to  15 :  also  of  4  up  to 20. 


74  MENTAL    ARITHMETIC. 

What  one  of  the  numbers  just  named  is  a  multiple 
both  of  3  and  of  4? 

Any  product  which  has  two  or  more  numbers  as 
factors  of  it  is  a  common  multiple  of  those  numbers. 

Name  the  multiples  of  6  up  to  24,  and  of  9  up  to 
27,  and  state  what  common  multiple  you  find  of  6 
and  9. 

Name  the  multiples  of  8  up  to  32,  and  of  12  to  36, 
and  state  what  common  multiple  you  find  of  8  and  12. 

Name  all  the  multiples  of  6  and  of  8  below  the 
number  50.  How  many  common  multiples  have  you 
found  of  6  and  8? 

What  is  the  least  common  multiple  of  6  and  8  ? 

Find  a  common  multiple  of  6  and  9. 

Find  a  second  common  multiple  of  6  and  9. 

What  is  the  least  common  multiple  of  6  and  9  ? 

What  is  the  least  common  multiple  of  8  and  10  ? 

What  is  the  least  common  multiple  of  3,  4,  and  6  ? 

What  is  the  least  common  multiple  of  4,  6,  and  8  ? 

From  the  above  exercise  it  will  be  seen  that  the  least 
common  multiple  of  two  or  more  numbers  is  the  small- 
est product  that  contains  them  all  as  factors. 

The  least  common  multiple  will  always  be  the  first 
one  found  in  enumerating  the  series,  as  in  the  above 
examples. 

But  this  method  is  often  tedious ;  and  we  now  pro- 
ceed to  exhibit  another  method,  independent  of  any 
such  series. 

Suppose,  now,  we  wish  to  find  the  smallest  common 
multiple  of  3  and  5.  The  number,  it  is  clear,  must 
be  a  certain  number  of  3's,  and  also  a  certain  number 
of  5's.  Now,  by  multiplying  3  and  5  together,  we 
evidently  obtain  such  a  number ;  for  it  will  be  3  times 
5,  and  it  will  be  5  times  3.  Multiplying  the  two  num- 
bers together,  then,  will  always  give  their  common 
multiple.     The  next  question  is,  Will  this  product  of v 


FRACTIONS.  75 

the  two  numbers  be  their  least  common  multiple? 
This  will  depend  on  the  character  of  the  numbers. 
If  the  numbers  are  prime  to  each  other,  their  product 
will  be  their  least  common  multiple.  For  example,  in 
the  numbers  3  and  5,  if  we  take  any  number  of  5's  less 
than  3,  as  2X5,  the  factor  3  has  disappeared,  and  the 
number  is  no  longer  a  multiple  of  3.  If  we  take  any 
number  of  3's  less  than  5,  as  4X3,  the  factor  5  has 
disappeared,  and  the  number  is  no  longer  a  multiple  of 
5.  The  product,  therefore,  of  numbers  prime  to  each 
other,  is  their  least  common  multiple.  In  the  above 
example,  the  numbers  3  and  5  were  prime  in  them- 
selves, and  not  merely  prime  to  each  other.  To  make 
the  principle  more  clear,  we  will  take  two  numbers 
that  are  not  prime  in  themselves,  but  are  only  prime 
to  each  other. 

What  is  the  least  common  multiple  of  8  and  9? 
Multiplying  them  together,  we  have  72.  72  is,  then, 
a  common  multiple  of  .8  and  9.  The  question  is,  Is 
it  their  smallest  common  multiple?  Writing  the  num- 
bers with  their  factors,  they  are  2X2X2,  and  3X3. 
Now,  if  we  erase  one  of  the  2's,  we  have  no  longer 
the  factors  of  8,  and  the  product  of  the  factors  will 
not  be  divisible  by  8.  In  the  same  way,  if  we  erase 
one  of  the  3's,  the  product  will  not  be  divisible  by  9. 

If,  then,  the  numbers  are  either  prime,  or  prime  to 
each  other,  the  product  is  their  least  common  multiple. 

Next,  let  us  inquire,  What  is  the  least  common  mul- 
tiple of  4  and  6?  Their  product  is  24;  but  this  is 
evidently  not  their  least  common  multiple,  for  12  con- 
tains both  4  and  6  as  factors.  To  show  why  it  is, 
that,  in  this  case,  something  less  than  the  product  of 
the  numbers  is  their  least  common  multiple,  we  will 
express  each  by  its  factors,  thus,  2X2,  2X3.  Now,  it 
is  clear  that  any  number  of  times  which  you  take 
2X2  as  a  factor  will  be  a  multiple  of  2x2.  If,  then, 
we  throw  out  the  2  in  the  2X3,  and  multiply  by  the 


76  MENTAL,    ARITHMETIC. 

remaining  3.  the  product  will  be  a  multiple  of  2X2, 
or  4.  Looking,  now,  at  the  2X3,  or  6,  it  is  evident 
that  any  number  of  times  which  you  may  take  that 
as  a  factor  will  be  a  multiple  of  2X3.  But  the  2  we 
may  take  from  the  2X2,  throwing  away  that  in  the 
2X3:  this  leaves  us  to  multiply  the  2X3  by  2;  as 
we  before  multiplied  the  2X2  by  3,  making  12  as  the 
least  common  multiple.  The  *ule,  therefore,  is:  Re- 
tain of  each  prime  factor  the  highest  power  which* 
appears  in  any  of  the  given  numbers;  erase  the  rest, 
and  multiply  together  what  then  remain. 

Find  the  least  common  multiple  of  8,  24,  and  36. 
Expressed  by  the  factors,  they  are  2X2X2;  2X2X2 
X3;  2X2X3X3.  Now,  2X2X2  is  common  to  8  and 
24;  it  may  be  thrown  out  of  the  latter,  leaving  only  3. 
Examining  again,  you  observe  that  2X2  is  common 
to  8  and  36;  we  throw  this  out  of  36,  leaving  3X3. 
Finally,  3,  we  find,  is  common  to  24  and  36;  throw- 
ing this  out  of  24,  we  find  the  numbers  appear  as  fol- 
lows: 2X2X2;  2X£X£x£;  £X2X3X3.  These 
multiplied  together  give  for  the  least  common  multi- 
ple, 72.  This  conforms  to  the  rule;  for  2X2X2  is 
the  highest  power  of  the  factor  2,  and  3X3  of  the 
factor  3.  What  is  the  least  common  multiple  of  24, 
60,  and  100?  These  factors  are  2X2X2X3;  2X2X3 
X5;  2X2X5X5.  We  see  that  2X2  is  common  to 
them  all;  expunge  it  in  the  second  and  third  number. 
Next,  3  is  common  to  the  1st  and  2d ;  expunge  it  in 
the  2d.  Lastly,  5  is  common  to  the  2d  and  3d;  ex- 
punge it  in  the  2d,  and  the  numbers  will  stand,  2X2 
X2X3;  &X<*X$X$;  2X2X5X5.  These  multiplied 
together  give  600. 

To  multiply  these  most  easily,  first  take  2X2X5 
X5=10();  then  the  remaining  factors,  2X3,  mul- 
tiplied by  100,  give  600. 

What  is  the  least  common  multiple  of  24,  40, 
and  72? 


FRACTIONS.  77 

What  is  the  least  common  multiple  of  18,  54,  81  ? 
What  is  the  least  common  multiple  of  15,  4,  7?    Of 
15,  40,  27?    Of  16,  14,  6?    Of  60,  12,  18? 

From  the  foregoing  reasoning  and  examples,  you 
wiU  perceive  that  the  least  common  multiple  of  sev- 
eral numbers  is  the  product  of  all  their  prime  factors, 
each  taken  in  the  highest  power  in  which  it  appears 
in  any  of  the  numbers. 


SECTION    XI. 

PRACTICAL   QUESTIONS. 

1.  What  part  of  a  shilling  is  1  penny?  2  pence? 
3  pence  ?    4  pence  ?    5  pence  ?    6  pence  ?    7  pence  ? 

2.  What  part  of  a  penny  are  2  farthings?  3  far- 
things? 4  farthings?  5  farthings?  6  farthings?  8 
farthings  ? 

3.  What  part  of  a  shilling  is  1  farthing?  2  far- 
things?   3  farthings? 

What  part  of  a  shilling  is  1  penny  and  1  farthing  ? 
1  penny  2  farthings?  3d.  3qrs. ?  4d.  2qrs. ?  6d.  1 
qr.?     9d.  2qrs.? 

4.  What  part  of  a  pound  is  1  shilling?  2s.?  3s.? 
5s.?  Is.  Id.?  2s.  Id.?  4s.  3d.?  "5s.  6d.?  7s.  9d.? 
3s.  8d.? 

5.  What  part  of  a  pound  is  1  farthing  ?  2  qrs.  ? 
3  qrs.?  2d.  3  qrs.?  5d.  2  qrs.?  Is.  Id.  1  qr.  ?  6  s. 
7d.  3  qrs.? 

6.  What  part  of  a  pound  avoirdupois  is  2  oz.  ? 
3  oz.?  4oz.?  5  oz.  ?  6  oz.  ?  7  oz.  ?  8  oz.  ?  9  oz.  ? 
10  oz.? 

7.  What  part  of  one  ounce  is  one  dram  ?    What  part 

7* 


78  MENTAL    ARITHMETIC. 

of  one  pound  is  one  dram  ?  2  drs.  ?  3  drs.  ?  1  oz.  1 
dr.  ?  1  oz.  2  drs.  ?  2  oz.  4  drs.  ?  3  oz.  6  drs.  ?  8  oz. 
3  drs.?    9  oz.   11  drs.? 

8.  What  part  of  a  pound  is  TV  of  an  oz.  ?  T\  of 
an  oz.? 

What  part  of  a  pound  is  f  an  oz.  ?  2}  oz.  ?  3£ 
oz.?    4£  oz.  ? 

9.  What  part  of  a  pound  Troy  is  1  dwt.  ?  5  dwt.  ? 
6  dwt.  ?    9  dwt.  ?    1 1  dwt.  ?    10  dwt.  ?     1  oz.  1  dwt.  ? 

3  oz.  4  dwt.  ? 

What  part  of  an  oz.  Troy  is  1  dwt.  ?    3  dwt.  1  gr.  ? 

4  dwt.  6  gr.  ?  7  dwt.  3  gr.  ?  8  dwt.  9  grs.  ?  10  dwt.  ? 
12  dwt.?     16  dwt.  ? 

10.  What  part  of  an  ell  English  is  1  qr.  of  a 
yard?  2  errs.  ?  3  qrs.  ?  What  part  of  a  qr.  is  1 
nail?    3  nails? 

11.  What  part  of  a  yd.  is  1  qr.  1  nail  ?  2  qrs.  3  n.  ? 
3  qrs.  2  n.?    What  part  of  an  ell  English  is  3  nails? 

1  qr.  3  n.  ?    4  qrs.  In.? 

12.  What  part  of  a  yd.  is  1  inch  ?  4  inches  ?  7 
inches  ?    9  inches  ?    What  part  of  a  yard  is  1  qr.  2  in.  ? 

2  qrs.  3  in.  ?    3  qrs.  1  in.  ? 

13.  From  a  vessel  containing  3  gallons  of  wine  3 
gills  leaked  out.  What  part  of  a  gallon  leaked  out  ? 
What  part  of  a  gallon  remained? 

14.  From  a  barrel  full  of  wine  7  quarts  were  drawn. 
How  many  quarts  remained?  What  part  of  the  barrel 
had  been  drawn  out?  What  part  of  the  barrel  had 
remained  ? 

15.  If  f  of  a  barrel  of  beer  be  divided  into  4  equal 
parts,  what  part  of  a  barrel  will  each  of  the  parts  be? 
How  many  gallons  will  each  part  be  ? 

16.  If  one  quart  be  taken  from  a  barrel  full  of  beer, 
what  part  of  a  barrel  will  remain?  If  3  pints  be 
taken  out,  what  part  will  remain?  If  7£  gallons  be 
taken  out,  what  part  of  a  barrel  is  taken  out?  What 
part  of  a  barrel  remains  ? 


FRACTIONS.  79 

17.  A  man  distributed  7£  gallons  of  milk  among  5 
persons.     What  part  of  a  gallon  did  he  give  to  each  ? 

18.  If  you  have  3^-.  gallons  of  milk,  and  distribute 
it  to  some  poor  persons,  giving  §  of  a  gallon  to  each, 
how  many  persons  will  you  give  it  to  ?  How  much 
will  remain? 

19.  What  part  of  1  foot  is  H  in.  ?  24  in.  ?  5£  in.  ? 
62-  in.  ?    8|  in.  ?    9±  in.  ?     10§  in.  ?     1 1*  in.  ? 

20.  What  part  of  a  yard  is  2  inches?  34  inches? 
14  in.?    5^  in.?    6±in.?     17^  in.?    24^  in.?" 

21.  What  part  of  a  rod  is  £  a  foot?  14  feet?  24 
feet  ?    4  ft.  3  in.  ?    6  ft.  7  in.  ?     10  ft.  5  in.  ? 

22.  What  part  of  3  rods  is  J-  a  foot?  1  foot?  3£ 
feet?    What  part  of  a  furlong  are  2£  rods?    5£  rods? 

23.  What  fraction  of  a  foot  is  f  of  a  yard?  f  of  a 
yd.  ?  What  fraction  of  a  foot  is  TV  of  a  rod?  T2T  of  a 
rod  ?    T33  of  a  rod  ? 

24.  A  man  measured  the  length  of  his  barn  with  a 
stick  half  a  yard  long,  and  found  the  barn  314  times 
the  length  of  his  stick.     How  long  was  it? 

25.  A  carpenter  is  cutting  up  a  board,  17i  feet  in 
length,  into  pieces  2£  feet  long.  How  many  pieces 
will  there  be,  and  how  long  will  be  the  piece  that 
remains? 

26.  A  man  measures  a  piece  of  fence  with  a  pole 
9£  feet  long.  The  fence  is  15J-  times  the  length  of 
the  pole.     How  many  rods  is  it  in  length? 

27.  What  part  of  a  peck  is  ^  of  a  bushel  ? 

What  part  of  a  gallon  are  TV  of  a  peck  ?    f  of  a  peck  ? 
What  part  of  a  quart  is  ^V  of  a  peck  ?    -gs  of  a  peck  ? 
What  part  of  a  quart  are  T\  of  a  bushel  ?  A  of  a 
bushel  ? 

28.  What  part  of  a  peck  is  |of  a  bush.  ?  f  of  a 
bush.  ?  |  of  a  bush.  ?  f  of  a  bush.  ?  f  of  a  bush.  ? 

29.  Two  men  bought  a  lot  of  standing  wood  in 
company,  for    11    dollars.     One  cut  off  2  cords,  the 

'Other  1  cord.     What  ought  each  to  pay  ? 


80  MENTAL    ARITHMETIC. 

30.  Two  boys  bought  the  chestnuts  on  a  tree  for 
50  cents.  One  had  11  quarts,  the  other  6  quarts  and 
1  pint.     What  ought  each  to  pay  ? 

31.  Three  men  bought  a  piece  of  cloth  for  24  dol- 
lars. The  first  took  2£  yds.,  the  second  the  same 
quantity ;  and  on  measuring  the  remainder,  it  was 
found  to  be  3  yards.      What  ought  each  to  pay  ? 

32.  Two  men  hire  a  horse  for  a  month  for  12  dol- 
lars. One  travels  200  miles  with  the  horse,  the  other 
150.     How  much  should  each  pay  ? 


SECTION    XII. 

DECIMAL    FRACTIONS. 

[See  Numeration,  Part  II.] 

In  the  calculations  in  common  fractions,  a  great  in- 
convenience arises  from  their  irregularity.  There  is 
no  law  regulating  the  magnitude  of  either  of  the 
terms.  The  denominator  may  be  in  any  ratio  what- 
ever to  the  numerator.  From  seeing  one  you  can 
make  no  inference  at  all  respecting  the  magnitude  of 
the  other.  In  calculations  of  addition,  it  is  often  more 
than  half  the  work  to  bring  the  fractions  into  a  com- 
mon denomination. 

Now.  it  is  evident,  that,  if  fractions  could  be  writ- 
ten in  the  same  manner  as  whole  numbers,  that  is, 
increasing  in  a  tenfold  rate  as  you  advance  to  the  left,a 
and  decreasing  in  a  tenfold  rate  as  you  advance  to  the 
right,  an  immense  gain  would  be  made  in  the  con- 
venience of  calculating  them.  Operations  in  fractions 
would  then  be  just  as  easy  as  operations  in  whole 
"lumbers.  Now,  this  advantage  is  gained  in  decimal 
fractions.     They  are  brought  under  the  same  law  asv 


DECIMAL    FRACTIONS.  81 

whole  numbers.  Let  us  observe  the  manner  in  which 
whole  numbers  are  written.  Take  the  number  222  ; 
the  right-hand  figure  signifies  two  units,  the  next  two 
tens,  the  next  two  hundreds ;  just  as  if  it  were  writ- 
ten in  this  manner,  2X100  +  2X10  +  2;  two  multi- 
plied by  100,  plus  two  multiplied  by  ten,  plus  two; 
making  two  hundred  and  twenty-two.  But  this  cum- 
bersome method  of  writing  is  unnecessary,  because 
the  law  of  notation  determines  what  number  the 
figures  in  each  place  shall  be  multiplied  by.  It  must 
not  be  forgotten  that  the  figure  2,  in  the  above  exam- 
ple, in  no  case  signifies  of  itself  more  than  two.  It  is 
the  place  it  occupies  that  gives  it  the  higher  value  of 
tens  or  hundreds. 

Now,  it  would  evidently  be  a  great  convenience  if 
we  could  reduce  fractions  to  the  same  law,  so  that 
they  would,  like  whole  numbers,  decrease  in  a  deci- 
mal ratio,  in  advancing  from  the  left  to  the  right.  To 
show  this  regularity  to  the  eye,  we  will  write  the  fol- 
lowing numbers:  two  multiplied  by  1000,  two  multi- 
plied by  100,  two  multiplied  by  10,  two  units,  two 
divided  by  10,  two  divided  by  100,  and  two  divided 
by  1000.  Written  in  full,  they  would  stand  thus : 
2X1000  + 2X100 +  2X10  +  2  + A +  T^  +  To2^. 

But  we  have  seen  that  we  may  write  the  whole 
numbers  without  the  multipliers,  thus,  2222,  because 
we  know,  from  the  place  each  figure  occupies,  what  its 
multiplier  must  be.  Just  so  we  can  write  fractions 
without  the  denominators,  provided  we  know,  from 
the  place  of  the  numerator,  what  the  denominator 
must  be.  Thus  the  whole  of  the  above  series  may 
be  written  as  follows :  2222.222.  A  decimal,  there- 
fore, is  the  numerator  of  a  fraction,  whose  denomina- 
tor is  never  written,  but  is  always  understood  to  be  1 
with  as  many  ciphers  as  there  are  places  in  the 
decimal. 

'      You  observe  that,  in  writing  the  series  given  above, 

T 


82  MENTAL,    ARITHMETIC. 

there  is  a  period  placed  at  the  right  hand  of  the  whole 
numbers,  separating  the  unit  figure  from  that  of 
tenths.  The  period  must  never  be  omitted  when 
there  are  fractions,  for  it  enables  you  to  determine  the 
value  of  each  figure  in  the  sum.  Instead  of  reading 
.22  two  tenths  and  2  hundredths,  we  may  call  it  22 
hundredths,  which  is  more  convenient,  and  amounts  to 
the  same  ;  for  two  tenths  is  equal  to  20  hundredths  ;  so 
.222  is  two  hundred  and  twenty-two  thousandths. 
So,  in  all  cases,  read  the  decimal  numbers  as  whole 
numbers,  and  for  their  denominator  take  1  with  as 
many  ciphers  as  there  are  places  in  the  written  deci- 
mals. 

In  all  your  study  of  decimals,  be  careful  not  to  con- 
found the  words  which  express  fractions  with  the 
similar  words  which  express  whole  numbers :  as  tenths 
with  tens,  hundredths  with  hundreds.  The  following 
questions  will  aid  you  in  fixing  this  distinction  clearly 
in  mind : 

1.  How  many  tenths  are  equal  to  ten  whole 
ones? 

2.  How  many  tenths  are  equal  to  two  and  a  half 
whole  ones  ? 

3.  How  many  hundredths  are  equal  to  three  and  a 
quarter  whole  ones? 

4.  How  many  hundredths  are  equal  to  one  hundred 
whole  ones? 

5.  How  many  thousandths  are  equal  to  ten  whole 
ones  ? 

6.  In  fifteen  whole  ones  how  many  tenths  ?  How 
many  hundredths  ? 

7.  In  seventy-five  hundredths  how  many  tenths  ? 

8.  In  three  tenths  how  many  hundredths? 

9.  hi  six  tenths  how  many  thousandths? 

Thus,  you  observe,  fractions  have  been  brought 
under  the  same  law  that  regulates  the  writing  of 
whole    numbers.     They   may   now   be   added,   sub- 


ADDITION    AND    SUBTRACTION    OF    DECIMALS.  83 

tracted,  multiplied,  and  divided,  like  whole  numbers. 
But.  in  doing  this,  it  is  important  to  determine  the 
place  of  the  period  that  separates  the  whole  numbers 
from  the  fractional  part  of  the  sum.  Where  must  the 
period  be  placed  in  the  answer  ? 


ADDITION    AND    SUBTRACTION    OF    DECIMALS. 

Let  us  first  observe  how  important  it  is  that  the 
rule  in  this  case  be  entirely  correct.  If  I  have  this 
number,  32.5,  to  write,  and  by  any  mistake  I  should 
write  it  3.25,  it  would  denote  a  quantity  only  one 
tenth  as  great  as  it  should  be;  or,  if  I  should  write 
325.  it  would  denote  a  quantity  ten  times  greater  than 
it  should  be.  Moving  the  period  one  place  to  the 
right,  makes  the  number  ten  times  as  great  as  it  was 
before ;  for  tens  become  hundreds,  and  hundreds, 
thousands ;  and  each  figure  ten  times  as  great  as  be- 
fore. Sor  by  moving  the  period  one  place  to  the  left, 
the  number  becomes  just  one  tenth  what  it  was  be- 
fore. Removing  the  period  two  places  from  its  true 
place,  makes  the  number  100  times  larger  or  smaller 
than  it  should  be,  according  as  you  remove  it  to  the 
right  or  the  left.  Hence  you  may  see  that,  in  order  to 
multiply  a  number  that  has  decimals,  by  10,  you  have 
only  to  remove  the  period  one  place  to  the  right  ;  to 
multiply  by  100,  remove  it  two  places ;  and  so  on. 
To  divide  by  10,  remove  the  period  one  place  to  the 
left :  to  divide  by  100,  remove  it  two  places ;  and  so 
on.  From  the  above  you  may  see  the  importance  of 
being  perfectly  accurate  in  fixing  the  place  of  the 
decimal  in   the  answer  to  any  question. 

We  will  begin   with  addition.     Add  4.46  to  3.21. 

Here  you  observe  the  two  whole  numbers  make  7, 

and  46  hundredths  added  to  21  hundredths  make  67 

•  hundredths.     The  answer,  then,  must  be  7.67,  having 


84  MENTAL    ARITHMETIC. 

two  (Jgcimal  places.  Add  6.8  to  5.23.  The  3  hun- 
dredths must  evidently  stand  alone,  since  there  is  noth- 
ing like  it  to  add  to  it ;  2  tenths  added  to  8  tenths 
make  10  tenths,  or  one  whole  one  ;  this  we  carry  to 
the  5,  which  gives  us  for  the  answer,  12.03.  This 
will  serve  to  suggest  the  rule  for  placing  the  period  in 
the  answer  to  questions  in  addition.  The  number  of 
decimal  places  in  the  answer  must  be  as  great  as  can 
be  found  in  any  one  of  the  numbers  to  be  added. 

The  same  rule  holds  in  subtraction.  Take  for 
illustration  the  numbers  given  in  the  second  example 
of  addition.  From  6.8  subtract  5.23.  Now,  as  in 
the  minuend  there  are  no  hundredths,  we  must  borrow 
10  in  this  place,  and  we  shall  have  a  remainder  of  7 
hundredths ;  adding  1  tenth  to  the  subtrahend,  to  com- 
pensate for  the  10  hundredths  added  to  the  minuend, 
we  have,  in  the  place  of  tenths,  a  remainder  of  5  ; 
finally,  in  the  place  of  units  we  subtract  5  from  6 ; 
the  answer  is  1.57.  In  performing  this  operation,  you 
may,  if  you  please,  call  the  8  tenths  80  hundredths  ; 
then  23  hundredths  from  80  hundredths  leaves  57  hun- 
dredths. By  performing  slowly  and  with  care  exam- 
ples of  your  own  selection,  you  will  see  the  verifica- 
tion of  the  rule  given  above,  both  for  addition  and 
subtraction. 

Add  2.4  to  3.8.  Add  .6  to  1.3.  Add  .4  to  .3. 
Add  .37  to  .25.     Add  3.7  to  .24.     Add  1.08  to  .05. 

From  4.6  subtract  2.4.  From  7.1  subtract  6.4. 
From  .18  subtract  .13.     From  4.5  subtract  .6. 

In  these  examples,  each  step  should  be  explained  by 
the  pupil  as  he  performs  it. 


MULTIPLICATION    OF    DECIMALS. 

The  rule  in  multiplication  we  shall  find  to  be  differ- 
ent from  the  above. 


MULTIPLICATION    OF    DECIMALS.  85 

1.  First,  we  will  multiply  2.4  by  3.  If  we  regard 
the  multiplicand  as  a  whole  number,  the  answer  will 
be  72.  But  by  regarding  the  multiplicand  as  a  whole 
number,  —  as  24  instead  of  2  and  4  tenths,  —  we  re- 
garded it  as  ten  times  greater  than  it  really  is.  The 
answer,  therefore,  is  ten  times  too  great.  Instead  of 
72,  it  must  be  7.2. 

2.  Multiply  6.2  by  3.4.  By  regarding  both  as 
whole  numbers,  we  obtain  the  answer  2108.  Now,  in 
calling  the  multiplicand  62  instead  of  6.2,  we  treated 
it  as  10  times  greater  than  it  is.  The  answer  must 
therefore  be  10  times  too  great,  even  if  the  multiplier 
were  a  whole  number.  We  must  therefore  divide  it 
by  10,  or  write  210.8.  But  the  multiplier  also  is  10 
times  too  great ;  the  answer  must  therefore  be  divided 
again  by  10,  in  order  to  bring  it  right.  Thus  the  an- 
swer will  stand  21.08. 

3.  Again ;  multiply  .62  by  3.4.  Here  we  obtain 
the  same  figures  as  before,  2108 ;  but,  by  treating  the 
multiplicand  as  a  whole  number,  we  regarded  it  as 
100  times  too  great ;  the  answer,  therefore,  must  be 
divided  by  100,  or  written  21.08.  But  the  multiplier, 
calling  it  a  whole  number,  was  taken  10  times  greater 
than  it  is ;  the  answer  must  be  again  divided  by  10, 
and  thus  it  will  stand  2.108. 

4.  Once  more ;  multiply  .62  by  .34.  The  figures 
of  the  answer  are,  as  before,  2108 ;  but,  by  regarding 
both  the  factors  as  whole  numbers,  we  take  each  100 
times  greater  than  it  is.  We  must  therefore  divide  by 
100  to  correct  the  error  in  the  multiplier,  and  again 
by  100  to  correct  the  error  in  the  multiplicand.  This 
will  remove  the  point  four  places  to  the  left,  and  the 
true  answer  will  be  .2108.  By  examining  these  ex- 
amples, you  will  see  that  the  pointing  in  each  case 
conforms  to  the  following  rule  : 

Point  off  as  many  figures  for  decimals  in  the  answer 
8 


86  MENTAL    ARITHMETIC. 

as  there  are  decimal  places  in  both  the  factors  taken 
together. 

5.  Multiply  2.7  by  .3.-6.  Multiply  .6  by  .7.-7. 
Multiply  6.  by  .7.-8.  Multiply  .02  by  .3.-9.  Multi- 
ply .02  by  .03.-10.  Multiply  .01  by ".01. 


DIVISION    OF    DECIMALS. 

1.  Divide  48  by  12.     Ans.  4. 

2.  Divide  4.8  by  12.  The  figure  expressing  the 
answer  is  4,  as  in  the  first  case ;  but,  observe,  the  divi- 
dend is  only  one  tenth  as  large  as  before  ;  the  quo- 
tient, therefore,  is  only  one  tenth  as  large.  Instead  of 
4.  it  is  .4. 

3.  Divide  .48  by  12.  The  figure  of  the  quotient 
is  still  4;  but,  as  the  dividend  is  only  one  hundredth 
part  as  large  as  in  the  first  example,  the  quotient  will- 
be  only  one  hundredth  part  of  4,  or  4  hundredths, 
written  thus,  .04. 

4.  Again  ;  divide  48  by  1.2.  The  quotient  is  still 
4;  but  we  must  investigate  the  question  to  see  where 
this  4  must  stand.  You  observe  that  the  divisor  is 
now  only  one  tenth  of  12.  Now,  if  the  divisor  is 
only  one  tenth  as  great  as  it  was  before,  you  must 
consider  how  that  will  affect  the  quotient.  You  will 
perceive,  on  reflection,  that,  as  you  diminish  the  divisor, 
you  increase  the  quotient.  If  you  make  the  divisor 
half  as  great,  the  quotient  will  be  twice  as  great ;  and 
so  proportionally  of  other  numbers.  Now,  as,  in  this 
instance,  the  divisor  is  one  tenth  as  great  as  before, 
the  quotient  must  be  ten  times  greater.  The  figure 
4,  then,  which  is  the  quotient  figure,  instead  of  stand- 
ing in  the  place  of  units,  as  before,  must  stand  in  the 
place  of  tens-;  that  is,  it  must  be  40,  the  cipher 
merely  showing  that  the  4  stands  in  the  piace  of  tens. 


DIVISION    OF    DECIMALS.  87 

5.  Once  more;  divide  48  by  .12.  Here,  again, 
yon  have  4  for  the  quotient  figure,  for  yon  can  have 
no  other ;  but,  on  comparing  this  example  with  the 
first,  yon  perceive  the  divisor  is  only  one  hundredth 
part  as  great ;  the  quotient  must  therefore  be  one  hun- 
dred times  greater  ;  that  is,  it  is  400,  the  ciphers  merely 
removing  the  4  into  the  place  of  hundreds. 

On  examining  these  examples  carefully,  you  will 
see  that  each  answer  is  unquestionably  correct.  "  But 
by  what  rule,"  you  ask,  "  are  these  examples 
wrought  ?  "  They  are  not  wrought  by  rule,  but  by 
reasoning  on  the  numbers  themselves  ;  and  the  more 
you  habituate  yourself  to  reason  in  arithmetic,  the 
less  need  you  will  have  to  depend  on  rules. 

With  this  suggestion  I  will  now  state  a  rule,  which 
you  may  at  any  time  follow,  when  you  have  not  time 
to  look  into  the  reason  of  the  operation. 

There  must  be  as  many  decimals  in  the  quotient  as 
the  decimals  in  the  dividend  exceed  those  in  the  divi- 
sor. When  there  are  fewer  decimals  in  the  dividend 
than  there  are  in  the  divisor,  ciphers  must  be  added  so 
as  to  make  the  number  equal. 

We  will  now  review  the  foregoing  examples,  and 
observe  their  conformity  with  the  above  rule.  Exam- 
ple 1  has  no  decimals  in  the  divisor  or  the  dividend, 
therefore  none  in  the  quotient.  Ex.  2,  the  dividend 
has  one  decimal,  the  divisor  none  ;  the  quotient  has 
therefore  one.  Ex.  3,  the  dividend  has  two  decimals, 
the  divisor  none;  the  quotient  has  two.  Ex.  4,  the 
dividend  has  none,  the  divisor  one ;  there  must  then 
be  a  cipher  added  to  the  dividend,  and  then  the  quo- 
tient will  be  in  whole  numbers.  Ex.  5,  the  divi- 
dend has  none,  the  divisor  two ;  there  must  then  be 
two  ciphers  added,  and  then  the  quotient  will  be  in 
whole  numbers. 

6.  Divide  45  by  15.  Divide  4.5  by  15.  Divide  .45 
by  15.     Divide  45  by  1.5.     Divide  45  by  .15. 


88  MENTAL    ARITHMETIC. 

7.  Divide  66  by  11;  6.6  by  11;  .66  by  11;  66  by 
1.1;  66  by  .11. 

In  calculations  of  Federal  money,  cents  and  mills 
are  regarded  as  decimals ;  the  point,  therefore,  separa- 
ting the  whole  numbers  from  the  fractions  must  be 
placed  between  the  dollars  and  the  cents.  Thus,  24.00 
is  24  dols. ;  2.40  is  2  dols.  40  cts.  ;  0.24  is  24  cts. 

8.  A  man  divided  $24.00  among  3  men.  How 
much  did  each  receive? 

9.  A  man  divided  $2.40  among  3  men.  How 
much  did  each  receive?     Divide  2.4  by  3. 

10.  A  man  divided  $0.24  among  3  men.  How 
much  did  each  receive?     Divide  $0.24  by  3. 

11.  A  man  divided  36  dollars  among  4  persons. 
How  much  did  each  receive?     Divide  36  by  4. 

12.  A  man  divided  $3.60  among  4  persons;  How 
much  did  each  receive  ?    What  is  one  fourth  of  $3.60  ? 

13.  A  man  divided  $0.36  among  4  men.  How 
much  did  each  receive?     What  is  one  fourth  of  .36? 


SECTION    XIII. 

REDUCTION  OF  VULGAR  FRACTIONS  TO  DECIMALS. 

We  have  now  seen  that  decimal  fractions  have  this 
great  advantage  over  vulgar  fractions,  —  that  they  con- 
form to  the  same  law  of  notation  as  whole  numbers, 
and  may  be  added,  subtracted,  multiplied,  and  divided, 
in  the  same  manner,  and  with  the  same  ease,  as  whole 
numbers.  It  is  desirable,  therefore,  to  introduce  them 
in  a  great  many  cases  instead  of  vulgar  fractions.  The 
next  question  that  arises,  therefore,  is,  Can  a  vulgar 
fraction  be  changed  to  a  decimal  having  the  same 
value ;  and  how  can  it  be  done  ?     Take  the  fraction 


ANALYSIS    OF    DECIMALS.  89 

£ ;  we  wish  to  reduce  it  to  tenths,  or,  in  other  words, 
to  express  it  in  tenths.  Now,  we  can  change  any 
number  to  tenths  by  multiplying  it  by  10.  Thus,  3 
is  30  tenths,  4  is  40  tenths.  We  will  now  take  £,  and 
change  the  numerator,  1,  to  tenths,  and  it  will  stand 
r  1.0;  but  the  fraction  was  not  one,  but  one  half  of  one  ; 
1.0,  therefore,  is  twice  as  great  as  it  should  be ;  we 
must  divide  it,  therefore,  by  2,  that  is,  by  the  denomi- 
nator, and  it  will  be  .5.  To  reduce  a  vulgar  fraction, 
then,  to  a  decimal ;  add  a  cipher  to  the  numerator,  and 
divide  by  the  denominator.  If  one  cipher  is  not 
enough  to  render  the  division  complete,  add  more. 

Reduce  to  a  decimal  \.  Change  the  numerator 
to  tenths;  it  will  be  1.0;  but  the  quantity  to  be 
reduced  to  tenths  was  not  one,  but  one  fifth  of  one  ; 
1.0,  therefore,  is  5  times  greater  than  it  should  be ; 
dividing  by  5,  the  answer  is  .2. 

Reduce  to  a  decimal  the  fraction  §,  explaining  each 
step  in  the  operation. 

Reduce  to  a  decimal  the  fraction  f. 

Reduce  to  a  decimal  the  fraction  f. 

Reduce  to  a  decimal  the  fraction  £. 

Reduce  to  a  decimal  the  fraction  f. 

I  will  here  direct  your  attention  to  a  fact  that  it  is 
interesting  to  notice.  If  the  denominator  of  the  vul- 
gar fraction  is  one  of  the  factors  of  10,  that  is,  if  it  is 
either  2  or  5,  the  decimal  figure  will  be  as  many  times 
the  other  factor  as  there  are  units  in  the  numerator  of 
the  vulgar  fraction.  This  will  appear  self-evident 
i  when  we  express  the  numbers  by  their  factors.  Thus, 
in  obtaining  the  decimal  for  £,  we  divide  10  by  2;  but 
10  is  2X5 ;  therefore,  in  dividing  by  2,  we  simply  ex- 
punge the  factor  we  divide  by,  and  leave  the  other  ; 
2)^X5.  So  in  the  fraction  -£,  we  obtain  the  decimal 
by  dividing  10  by  5,  which  expunges  the  factor  5  ; 
,  5)#X2.  In  reducing  f,  we  divide  2X10  by  5,  thus, 
5)2X2X£,  leaving  twice  the  factor  2 :  in  f,  5)3X2 
8* 


90 


MENTAL    ARITHMETIC. 


X$,  leaving  3  times  the  factor  2;  in  f,  5  )  2X2X2X0, 
leaving  4  times  the  factor  2. 

2.  We  will  now  take  the  fraction  £.  Proceeding  as 
before,  we  wish  to  divide  10  by  4,  thus,  2X2)2X5. 
Here  we  see  the  division  cannot  be  complete ;  for  the 
divisor  contains  the  factor  2  twice,  while  the  dividend 
has  it  only  once.  If,  however,  we  had  multiplied 
the  original  numerator,  1,  by  100,  instead  of  10,  we 
should  have  had  10  twice  as  a  factor  in  the  dividend, 
and  of  course  each  factor  of  10  twice  ;  100  is  10X10, 
and  10  is  2X5.  It  would  have  stood  then  thus,  2X2  ) 
2X5X2X5.  The  division  is  now  complete;  for  the 
dividend  contains  the  factor  2  as  many  times  as  the 
divisor  has  it.  Expunging  these,  we  have  remaining 
the  factor  5  taken  twice,  or  .25. 

This  process,  you  may  observe,  conforms  to  the 
rule,  to  add  as  many  ciphers  as  may  be  necessary  to 
render  the  division  complete. 

3.  Reduce  the  vulgar  fraction  f  to  a  decimal.  30 
is  composed  of  the  prime  factors  3X2X5;  it  con- 
tains 2  only  once,  and  therefore  it  is  not  divisible  by 
2X2;  30  must  therefore  be  multiplied  by  10.  This 
will  introduce  another  2,  and  it  will  stand  thus,  2X2) 
3X2X5X2X5.  By  expunging  the  two  2's,  and  mul- 
tiplying together  the  other  factors,  we  have  .75  for  the 
answer. 

4.  Reduce  the  fraction  £  to  a  decimal.  10,  ex- 
pressed by  its  factors,  is  2X5,  and  8  is  2X2X2.  We 
must  therefore  multiply  2X5  by  10  till  it  shall  contain 
the  factor  2  as  many  times  as  8  contains  the  same  fac- 
tor ;  that  is,  the  numerator,  1,  must  be  multiplied  by 
a  thousand.  It  will  then  stand,  2X2X2)2X5X2X5 
X2X5.  Expunging  the  three  2's,  there  remains  for 
the  answer,  .125. 

By  examining  the  above  examples,  you  may  observe 
this  fact, — that  if  the  denominator  of  the  vulgar  fraction, 
contains  one  of  the  factors  of  10,  that  is,  2  or  5,  one 


ANALYSIS    OF    DECIMALS.  91 

or  more  times  as  a  factor,  the  decimal  will  contain  the 
other  factor  just  as  many  times.  Thus,  J=.5j  i  or 
^  =  .25,  or  .5X.5;  i  or  sxl**  =^-*25,  or  .5X.5X.5. 

In  the  same  way,  i=.2]  A  or  ^  =  .04,  or  .2X.2: 
t**  or  *xix5  =  -008,  or  .2X.2X.2.  In  this  way  you 
may  determine  that  TV,  when  reduced  to  a  decimal, 
will  contain  5  four  times  as  a  factor,  because  16  con- 
tains 2  four  times  as  a  factor.  So  ^V  will  contain  5 
five  times  as  a  factor. 

This  is  conveniently  expressed  by  saying,  whatever 
power  of  one  of  the  factors  of  10  the  denominator  of 
the  vulgar  fraction  contains,  the  same  power  of  the 
other  factor  will  .appear  in  the  decimal. 

5.  Reduce  £  to  a  decimal  fraction.  Preparing  the 
numbers  as  before,  it  will  stand  3)2X5.  You  ob- 
serve that  3  is  different  from  either  of  the  factors  of 
10.  Now,  as  10  has  only  the  factors  2  and  5,  it  is  not 
divisible  by  3  without  a  remainder. 

If  you  add  to  the  numerator  ever  so  many  ciphers, 
you  will  only  increase  the  number  of  times  that  2  and 
5  appear  in  it  as  its  factors,  and  the  number  can  never 
become  divisible  by  3  without  a  remainder.  The  an- 
swer becomes  .333+.  and  this  indefinitely,  as  far  as 
you  may  please  to  carry  on  the  operation.  On  the 
same  principle,  we  shall  find  that  it  is  not  possible  to 
express  accurately,  in  decimals,  any  vulgar  fraction 
whose  denominator  contains  as  a  factor  any  thing  dif- 
ferent from  the  factors  of  10;  for  this  denominator 
becomes,  in  the  reduction,  a  divisor  of  10  or  some 
power  of  10 ;  and  if  it  has  any  thing  in  it  as  a  factor 
which  is  prime  to  the  factors  of  10,  the  complete  di- 
vision is  impossible.  Thus  £  cannot  be  exactly  ex- 
pressed in  decimals ;  because,  though  one  of  its  factors, 
2,  is  a  divisor  of  10,  the  other,  3,  is  prime  to  10.  On 
this  principle  the  following  questions  may  be  examined. 

Can  |  be  accurately  expressed  in  decimals  ?     Why  ? 

Can  £  be  accurately  expressed  in  decimals?     Why? 


92  MENTAL    ARITHMETIC. 

Can  £  be  accurately  expressed  in  decimals?     Why? 
Can  TV  be  accurately  expressed  in  decimals  ?  Why  ? 
Can  ^ ?   TV?  A?  A?    A?   A*    A?   A?   A?    A? 
A?  A?  A?  A? 

6.  Name  all  the  denominators,  from  2  up  to  20,  of 
such  fractions  as  can  be  accurately  expressed  in  deci- 
mals. From  20  to  40.  From  40  to  60.  From  60  to 
80. 

7.  Name  all  the  denominators,  from  2  to  20,  of  such 
fractions  as  cannot  be  expressed  accurately  in  decimals. 
From  20  to  40.     From  40  to  60.     From  60  to  80. 

8.  What  is  the  value  of  4  shillings,  expressed  in  the 
decimal  of  a  £  ?  As  1  shilling  is  j&  of  a  £,  4  s.  is 
sV.  We  can  change  4  to  tenths  by  adding  a  cipher  ; 
it  will  then  be  40.  4,  however,  was  not  the  number 
we  wished  to  reduce  to  tenths,  but  /ff  5  the  answer, 
40,  is  therefore  20  times  too  great ;  dividing  by  20,  it 
stands  .2.     4  shillings,  then,  is  2  tenths  of  a  £. 

9.  Now,  reverse  the  operation.  What  is  the  value, 
in  shillings,  of  .2  of  a  £  ?  Now,  shillings  are  twen- 
tieths. We  can  change  any  number  to  twentieths  by 
multiplying  it  by  20;  as,  1  is  20  twentieths,  2  is  40 
twentieths,  &c.  Multiplying  the  .2  by  20,  we  have 
40;  but  observe  the  2  was  not  two  wholes,  but  two 
tenths;  the  answer,  40,  therefore,  is  ten  times  too 
great.     Dividing  by  10,  the  answer  is  4  shillings. 

10.  Reduce  to  the  decimal  of  a  £,  2  shillings.  5 
shillings. 

11.  What  is  the  value,  in  shillings,  of  .1  of  a  £  ? 
Of  .25  of  a  £  ? 

12.  Reduce  to  the  decimal  of  a  shilling,  3  pence.  3 
pence  are  f?  of  a  shilling.  Reducing  to  hundredths, 
to  render  the  division  complete,  the  answer  is  .25. 

13.  What  is  the  value,  in  pence,  of  .25  of  a  shilling  ? 

14.  Reduce  9  pence  to  the  decimal  of  a  shilling. 

15.  Reduce  1  peck  to  the  decimal  of  a  bushel. 

16.  Reduce  3  pecks  to  the  decimal  of  a  bushel. 


INTEREST.  93 

17.  Reduce  .5  of  a  bushel  to  pecks.     .75  of  a  bu 
to  pecks. 

18.  Reduce  15  minutes  to  the  decimal  of  an  hour. 

19.  Reduce  45  minutes  to#the  decimal  of  an  hour. 

20.  Reduce  to  minutes  .5  of  an  hour.  .25  of  an 
hour.     .75  of  an  hour. 

21.  Reduce  6  inches  to  the  decimal  of  a  foot. 
9  inches  to  the  decimal  of  a  foot.  3  inches  to  the 
decimal  of  a  foot. 


SECTION    XIV. 

INTEREST. 

Interest  is  the  sum  paid  by  the  borrower  to  the 
lender  for  the  use  of  money.  The  rate  of  interest 
is  established  by  law,  and  varies  in  different  countries. 
In  England,  it  is  5  per  cent.,  that  is,  5  for  the  use  of 
100  for  1  year ;  in  the  New  England  States,  it  is  6 
per  cent. ;  in  New  York,  it  is  7  per  cent.  When  no 
particular  rate  is  mentioned  in  this  book,  6  per  cent, 
will  be  understood. 

If  I  borrow  100  dollars  for  1  year,  at  the  end  of  the 
year  I  owe  the  sum  I  borrowed,  100  dollars,  and  6 
dollars  for  the  use  of  it,  making  106  dollars.  The 
sum  borrowed  is  the  principal;  the  sum  paid,  for  the 
use  of  it  is  the  interest;  the  principal  and  interest 
added  together  make  the  amount. 

1.  What  is  the  interest  of  100  dols.  for  2  years?  3 
years?    4  years?    5  years?    6  years?    7  years? 

2.  What  is  the  interest  of  200  dols.  for  2  years  ?  3 
years  ?    4  years  ?    5  years  ?    6.  years  ? 

3.  What  is  the  interest  of  300  dols.  for  2  years? 
For  4  years  ?    Of  400  dols.  for  3  years  ? 


94  MENTAL    ARITHMETIC. 

4.  What  is  the  interest  of  50  dols.  for  1  year?    For 

3  years  ?    Of  25  dols.  for  1  year  ?    2  years  ? 

5.  What  is  the  interest  of  100  dols.  for  1  year? 
What  is  the  interest  of^J.00  cents  for  1  year? 
What  is  the   interest   of  2  dols.  for  1  year?    Of  3 

dols.  ?  Of  4  dols.  ?  5  dols.  ?  6  dols.  ?  7  dols.  ?  8 
dols.  ?    9  dols.  ? 

6.  What  is  the  interest  of  36  dols.  for  1  year?  Of 
47  dols.  ?  Of  57  dols.  ?  Of  34  dols.  ?  Of  62  dols.  ? 
Of  89  dols.  ?  Of  125  dols.  ?  Of  136  dols.  ?  Of  207 
dols.?    Of  561  dots.? 

7.  What  is  the  interest  of  50  cents  for  1  year?  Of 
25  cents?  Of  10  cents?  Of  20  cents?  Of  30  cents? 
Of  40  cents?  Of  50  cents?  Of  70  cents?  Of  80 
cents?    Of  90  cents? 

8.  What  is  the  interest  of  50  dols.  60  cents  for  1 
year?    Of  84.30?    Of  96.40?    Of  112.25?    Of  230.75? 

9.  What  is  the  interest  of  100  dols.  for  6  months? 
For  3  months?    For  2  months?    For  1  month?    For 

4  months?  For  5  months?  For  7  months?  For  8 
months?  For  9  months?  For  10  months?  For  11 
months  ? 

10.  What  is  the  interest  of  10  dols.  for  6  mo.  ?  3 
mo.?  2  mo.?  1  mo.?  4  mo.?  5  mo.?  7  mo.?  8 
mo.?    9  mo.?    10  mo.?    11  mo.? 

11.  What  is  the  interest  of  1  dol.  for  6  mo. ?    1  mo.? 
The  interest  of  1  dollar  for  1  month  is  half  a  cent, 

and  for  any  number  of  months,  it  is  half  as  many 
cents. 

12.  What  is  the  interest  of  1  dollar  for  5  mo.  ?  7 
mo.?  8  mo.?  9  mo.?  11  mo.?  12  mo.?  15  mo.? 
16  mo.?     17  mo.?     18  mo.? 

The  interest  of  any  number  of  dollars  for  1  month 
is  half  as  many  cents. 

13.  What  is  the  interest  o(  12  dollars  for  1  mo.  ? 
Of  15  dols.?  25  dols.?  37  dols.?  42  dols.?  67 dols.? 
93  dols.  ?    104  dols.  ? 


INTEREST.  95 

14.  What  is  the  interest  of  12  dols.  for  3  months? 
What  is  the  interest  of  25  dols.  for  6  months? 

In  computing  interest,  a  month  is  reckoned  30  days. 
As  the  interest  on  a  dollar  for  30  days  is  half  a  cent, 
that  is,  5  mills,  the  interest  on  a  dollar  for  1  fifth  of  30 
days  will  be  1  mill.  One  fifth  of  30  is  6;  the  in- 
terest, therefore,  on  1  dollar  for  6  days  is  1  mill ;  and 
the  interest  on  any  number  of  dollars  for  6  days  will 
be  as  many  mills  as  there  are  dollars. 

15.  What  is  the  interest  of  15  dollars' for  6  days? 
Of  25  dols.  ?  Of  40  dols.  ?  Of  65  dols.  ?  Of  75  dols.  ? 
Of  100  dols.  ?  Of  500  dols.  ?  Of  360  dols.  ?  Of  840 
dols.?    Of  1000  dols.? 

As  the  interest  of  1  dollar  for  6  days  is  1  mill,  for 
12  days  it  will  be  2  mills,  for  18  days  3  mills,  &c. 

16.  What  is  the  interest  of  1  dol.  for  24  days?  Of 
2  dols.  for  6  days?  Of  2  dols.  for  12  days?  Of  2  dols. 
for  18  -days?  Of  5  dols.  for  6  days?  For  12  days? 
For  24  days?    Of  36  dols.  for  18  days? 

17.  What  is  the  interest  of  125  dols.  for  1  year  and 

6  mo.? 

18.  What  is  the  interest  of  268  dols.  for  1  year?  For 
2  years?     For  3  years? 

19.  What  is  the  interest  of  45   dols.  for  4  years 

7  mo.? 

20.  What  is  the  interest  of  60  dols.  for  1  year  3 
mo.   18  days? 

21.  What  is  the  interest  of  100  dols.  for  2  years  1 
mo.   12  days? 

22.  What  is  the  interest  of  165  dols.  for  3  years  2 
mo.  6  days? 

23.  What  is  the  interest  of  50  dols.  for  one  month  1 
For  6  months  ?     For  1  vear  and  7  months  ? 


96 


MENTAL    ARITHMETIC. 


24.  What  is  the  interest  of  94  dols.  for  eight  mo.  24 
days? 

25.  What  is  the  interest  of  320  dols.  for  8  mo.  and  12 
days  ? 

26.  What  is  the  interest  of  84  dols.  for  4  mo.  and  15 
days? 

27.  What  is  the  interest  of  196  dols.  for  10  mo.? 

28.  What  is  the  interest  of  86  dols.  for  9  days? 

29.  What  is  the  interest  of  340  dols.  for  15  days? 

30.  What  is  the  interest  of  875  dols.  for  22  days? 

When  interest  is  more  or  less  than  6  per  cent.,  first 
find  the  interest  at  6  per  cent.,  and  then  make  a  pro- 
portional addition  or  subtraction  for  the  required  per 
cent.  If  it  is  7  per  cent.,  add  one  sixth;  if  5  per 
cent.,  subtract  one  sixth. 

31.  What  is  the  interest  of  140  dols.  for  1  year,  at 
7  per  cent.  ? 

32.  What  is  the  interest  of  200  dols.  for  1  year  and 
6  mo.,  at  5  per  cent.? 

33.  What  is  the  interest  of  460  dols.  for  1  year,  at 
A}  per  cent.  ? 

Remark.  —  4£  is  three  fourths  of  6. 

34.  What  is  the  interest  of  500  dols.  for  1  mo.,  at  9 
per  cent.? 


BANKING. 

When  money  is  obtained  at  a  bank,  the  note  which 
is  given  for  it  promises  to  pay  it  at  a  certain  time,  as 
60,  90,  or  120  days.  The  interest  on  this  note,  in- 
stead of  being  paid  at  the  end  of  the  time,  when  the 
note  is  taken  up,  is  paid  beforehand;  that  is,  it  is 
subtracted  from  the  sum  named  in  the  note;  so  that, 


BANKING.  97 

when  you  take  up  the  note,  you  have  only  to  pay  the 
face  of  it,  as  the  interest  has  been  paid  already. 

If  you  give  a  note  to  a  bank  for  100  dollars,  to  be 
paid  in  90  days,  they  subtract  from  the  sum  named  in 
the  note  the  interest  of  the  sum  for  90  days,  and  three 
days  besides,  called  days  of  grace;  the  balance  is  the 
sum  you  receive.  The  interest  of  100  dollars  for  90 
days  is  $1.50;  for  3  days,  it  is  5  cents.  $1.55  sub- 
tracted from  $100.00  leaves  a  balance  of  $98.45, 
which  is  the  sum  you  will  receive. 

If  the  note  is  given  for  60  days,  the  interest  is  cast 
for  63  days,  and  subtracted  from  the  sum  named. 

The  interest,  thus  subtracted,  is  called  the  bank 
discount;  and  the  bank,  when  it  lends  money  on  such 
a  note,  is  said  to  discount  the  note. 

35.  What  is  the  bank  discount  on  a  note  of  100 
dollars,  payable  in  30  days?  And  how  much  will  be 
received  on  such  a  note  ? 

The  interest  on  100  dollars  for  30  days  is  50  cents; 
for  3  days,  it  is  5  cents;  the  discount.  55  cents,  sub- 
tracted from  100  dollars,  leaves  $99.45,  the  sum  re- 
ceived. 

36.  What  is  the  bank  discount  on  a  note  for  200 
dollars  for  60  days?  And  what  is  the  cash  value  of 
the  note? 

37.  What  is  the  bank  discount,  and  what  is  the 
cash  value  of  a  note  for  150  dollars  payable  in  30 
days? 

38.  What  is  the  bank  discount,  and  what  is  the 
cash  value  of  a  note  for  200  dollars  payable  in  90 
days? 

39.  What  is  the  bank  discount,  and  what  is  the 
cash  value  of  a  note  for  300  dollars  payable  in  90 
days? 

9  G 


98  MENTAL    ARITHMETIC. 


DISCOUNT. 


When  money  is  paid  by  the  debtor  before  it  be- 
comes due,  an  allowance  is  made,  which  is  called 
f  discount.  If  I  owe  100  dollars,  to  be  paid  in  three 
months  from  this  time,  and  I  pay  it  now,  I  ought  not 
to  pay  the  full  hundred  dollars,  for  I  am  entitled  to  the 
use  of  the  money  three  months  longer.  The  sum  which 
should  be  paid  now,  to  cancel  a  debt  due  at  some 
future  time,  is  called  the  present  ivorth  of  the  debt. 

To  find  the  present  worth  of  a  debt  due  at  some 
future  time,  first  find  the  interest  on  the  debt  from  the 
time  of  payment  to  the  time  when  the  debt  is  due; 
subtract  this  interest  from  the  debt,  and  the  remainder 
will  be  the  present  worth.  Thus,  if  I  pay  a  debt  of 
100  dollars  three  months  before  it  is  due,  I  subtract 
the  interest  of  100  dollars  for  three  months  (=  $1.50) 
from  100  dollars,  leaving  $98.50  for  the  sum  which  I 
must  pay. 

This  rule  is  not  strictly  equitable,  because  $98.50, 
with  three  months'  interest  added,  will  not  amount  to 
$  100.  The  above  method,  therefore,  gives  the  present 
worth  a  little  too  small ;  but  it  is  the  method  uniformly 
adopted  in  business,  and  the  error  is  on  the  right  side, 
for  it  encourages  the  debtor  to  be  prompt  in  his  pay- 
ments. 

40.  What  is  the  present  worth  of  200  dollars  paya- 
ble in  1  year? 

41.  What  is  the  present  worth  of  150  dollars  paya- 
ble in  2  years? 

42.  What  is  the  present  worth  of  60  dollars  paya- 
ble in  6  months? 

43.  What  is  the  present  worth  of  530  dollars  paya- 
ble in  1  year? 

What  is  the  present  worth  of  400  dollars  payable  in 
1  year  and  6  months? 


LOSS    AND    GAIN.  99 


LOSS   AND    GAIN.— PER   CENTAGE. 

44.  A  boy  bought  a  penknife  for  25  cents,  and  sold 
it  for  28  cents.  How  many  cents  did  he  gain  on  a 
quarter  of  a  dollar? 

45.  Suppose  he  had  bought  4  knives  at  the  same 
price  each,  and  sold  them  at  the  same  profit,  he  would 
then  have  traded  with  a  dollar.  How  much  would  he 
have  gained  on  a  dollar? 

This  is  called  so  much  per  ce?it.,  which  only  means 
so  much  on  a  hundred. 

46.  A  boy  bought  a  bushel  of  apples  for  50  cents, 
and  sold  them  for  59  cents.  How  much  did  he  gain 
per  cent.  ? 

47.  A  bookseller  bought  a  book  for  75  cents,  and 
sold  it  for  84  cents.  How  much  did  he  make  per 
cent.  ? 

As  75  is  £  of  100,  what  he  gained  on  the  book  will 
be  £  of  what  he  would  gain  on  a  hundred,  or  what  he 
would  gain  per  cent. 

48.  A  boy  bought  some  melons  for  40  cents,  and 
sold  them  for  60  cents.    What  did  he  make  per  cent.  ? 

Ans.     His  gain  was  equal  to  half  his  outlay. 

49.  A  grocer  bought  a  lot  of  flour  for  5  dollars  a 
barrel ;  but,  finding  it  damaged,  he  sold  it  for  4  dollar? 
a  barrel.     What  did  he  lose  per  cent.  ? 

50.  A  man  bought  a  share  in  a  bank  for  80  dollars ; 
and  sold  it  for  82  dollars.    What  did  he  gain  per  cent.  ? 

51.  A  man  bought  a  lot  of  apples  for  $1.50  a  bar- 
rel.   What  must  he  sell  them  for  to  gain  10  per  cent.  ? 

52.  A  hatter  bought  some  hats  for  $3.50  each.  He 
is  willing  to  sell  them  at  a  profit  of  4  per  cent.  At 
what  price  will  he  sell  them  ? 

53.  A  manufacturing  company  declare  a  dividend  of 
7i  per  cent.  What  ought  a  stockholder  to  receive 
who  owns  350  dollars  in  that  factory? 


100  MENTAL    ARITHMETIC. 

54.  A  has  a  note  against  B  for  140  dollars,  which 
he  sells  for  cash  at  4  per  cent,  discount.  What  does 
he  receive  for  the  note? 

55.  A  merchant  buys  100  barrels  of  flour  for  5 
dollars  a  barrel,  and  sells  it  so  as  to  lose  5  per  cent. 
What  does  he  sell  it  for  a  barrel  ? 

He  afterwards  buys  250  casks  of  lime  at  1  dollar  a 
cask.  He  wishes  to  sell  it  so  as  to  make  good  his 
loss  on  the  flour.  At  what  per  cent,  profit  must  he  sell 
it,  and  for  how  much  a  cask  ? 

You  observe  that  the  money  invested  in  lime  is 
only  one  half  as  much  as  was  invested  in  flour. 

56.  A  lends  B  10  dollars  for  2  months,  without  in- 
terest. Afterwards  B  lends  A  5  dollars.  How  long 
can  A  keep  it  to  balance  the  favor  he  did  to  B  ? 

57.  O  lends  D  100  dollars,  without  interest,  for  4 
months.  Afterwards  D  lends  C  25  dollars.  How 
long  can  C  keep  it  to  balance  the  favor? 

In  these  cases,  you  will  see  that  the  money,  mul- 
tiplied by  the  time  it  was  kept,  must,  in  the  two  cases, 
be  equal.  If  10  dollars  is  lent  me  by  A,  without  in- 
terest, for  6  months,  I  can  balance  the  favor  by  lend- 
ing A  5  dollars  for  12  months,  or  4  dollars  for  15 
months,  or  15  dollars  for  4  months,  or  30  dollars  for  2 
months,  or  2  dollars  for  30  months,  or  20  dollars  for  3 
months. 

58.  A  lends  B  60  dollars  for  three  months,  without 
requiring  interest.  Afterwards  B  lends  A  90  dollars. 
How  long  may  A  keep  the  money  to  balance  the 
favor  ? 

59.  A  lends  B  40  dollars  for  three  months.  After- 
wards B  lends  to  A,  for  two  months,  a  certain  sum, 
the  use  of  which  should  balance  the  favor.  How 
large  must  the  sum  be? 

60.  A  lends  B  150  dollars  for  4  months.  B  after- 
wards lends  A  100  dollars.  How  long  can  A  keep  it. 
to  balance  the  favor  ? 


SQUARE    MEASURE. 


ioi 


SECTION    XV. 


SQUARE  MEASURE. 

Linear  measure  is  measure  in  a  straight  line,  having 
length  only.  Square  measure  is  the  measure  of  sur- 
face, having  length  and  breadth. 

Thus,  a  linear  inch,    ■ 


A  square  inch. 


*A  line   of  2  inches,  when  square,  will  therefore 
make  4  square  inches  j  thus, 


1.  A  line  of  three  inches, 
how  many  square  inches  ? 


when  square,  will  make 


Note  6. 


9* 


iu% 


MENTAL    ARITHMETIC. 


square 
Of  7? 


2.;  The    square,  of  .4  inches  is  how  many- 
inches  ?     The    square   of   5   inches  ?     Of   6  ? 
Of  8?     Of  9?     Of  10?     Of  11?     Of  12? 

3.  How  many  square  inches  are  there  in  a  square 
foot? 

4.  How  many  linear  feet  are  there  in  a  linear  yard  ? 
How  many  square  feet  in  a  square  yard  ? 

5.  How  many  square  inches  are  there  in  a  piece  of 
board  12  inches  long  and  3  inches  wide  ? 

6.  How  many  square  inches  are  there  in  a  piece  of 
board  8  inches  long  and  6  inches  wide  ?  In  a  board  5 
inches  long  and  3  inches  wide  ?  In  a  board  9  inches 
iong  and  5  inches  wide  ? 

7.  How  many  square  feet  are  there  in  the  floor  of  a 
room  12  feet  long  and  10  feet  wide  ? 

8.  How  many  square  feet  are  there  in  the  floor  of 
an  entry  15  feet  long  and  4  feet  wide  ? 

9.  How  many  square  yards  of  carpeting  will  cover 
a  room  6  yards  long  and  5  yards  wide  ? 

How  many  square  rods  are  there  in  a  piece  of  land 
14  rods  long  and  8  rods  wide  ? 

10.  If  a  road  4  rods  wide  passes  through  my  land 
for  the  distance  of  60  rods,  how  many  square  rods  of 
my  land  does  it  occupy  ? 

We  will  now  return  to  the  measure  of  an  inch.     If, 
instead  of   squaring  a  linear  inch, 
we    take    only  half  an    inch    and 
square  it,  we    shall  have  but  one 
fourth  of  a  square  inch  ;  thus, 


So  if  we  square  one  third  of  an 
inch,  it  will  give  us  $  of  a  square 
inch.  If  we  square  one  fourth  of 
an  inch,  it  will  give  us  TV  of  a  square  inch. 

11.  What  part  of  a  square  inch  will  £  of  an  inch, 
when  squared,  be  ?  What  part  of  a  square  inch  will 
£  of  an  inch  be  when  squared  ?    }  ?    i  ?   £  ?    TV  ?    tV  ? 

A* 


SQUARE    MEASURE.  103 

12.  What  part  of  a  square  inch  is  a  piece  of  paper 
1  inch  long  and  half  an  inch  wide  ?  One  inch  long 
and  three  fourths  of  an  inch  wide  ? 

13.  What  part  of  a  square  foot  is  a  board  1  foot  long 
and  half  a  foot  wide  ?  One  foot  long  and  9  inches 
wide  ? 

14.  How  many  square  inches  will  there  be  in  the 
square  of  a  line  1£  inches  long  ? 

[This  and  the  following  questions  may  be  answered 
by  drawing  the  figure  on  a  slate  or  on  a  board.] 

How  many  square  inches  will  there  be  in  the  square 
of  a  line  2^-  inches  long  ?  3£  ?  4£  ?  H  ?  6^  ?  7£  ?  8£? 
9^?   1(H? 

15.  How  many  square  feet  in  £  a  yard  squared  ? 

16.  How  many  square  inches  are  there  in  1  square 
foot? 

17.  How  many  square  feet  in  1  square  yard  ? 

18.  How  many  square  yards  in  1  square  rod  ? 

19.  How  many  square  feet  in  1  square  rod  ? 

40  sq.  rods  make  1  rood ;   4  roods  make  1  acre. 

20.  How  many  rods  make  1  acre  ? 

21.  If  a  piece  of  board  is  6  inches  wide,  how  long 
must  it  be  to  contain  a  square  foot  ? 

22.  If  a  piece  of  board  is  3  inches  wide,  how  long 
must  it  be  to  contain  a  square  foot  ? 

23.  How  long  must  it  be  to  contain  a  square  foot, 
if  it  is  2  inches  wide  ?  If  1  inch  wide  ?  If  4  inches 
wide  ? 

24.  If  cloth  is  £  a  yard  wide,  how  much  in  length 
will  make  a  square  yard  ? 

25.  How  much  lining  }  of  a  yard  wide  will  line 
one  yard  of  cloth  one  yard  wide  ? 

26.  If  cloth  is  f  of  a  yard  wide,  how  much  in 
length  will  it  take  for  a  square  yard  ? 


104  MENTAL    ARITHMETIC. 

27.  How  much  cloth  £  a  yard  wide  will  it  take  to 
line  7  yards  of  cloth  £■  of  a  yard  wide  ? 

How  much  -|  wide  will  line  1}  yards  f  wide  ? 

28.  How  long  must  a  strip  of  land  1  rod  wide  be 
to  contain  an  acre  ?  How  long,  if  2  rods  wide  ?  If 
3  rods  wide  ?  If  4  rods  wide  ?  If  8  rods  wide  ?  If 
10  rods  wide  ? 

29.  How  long  must  a  piece  of  land  be  to  contain  £ 
of  an  acre,  if  it  is  4  rods  wide  ? 

30.  If  a  piece  of  land  is  10  rods  in  length,  how 
wide  must  it  be  to  contain  £  an  acre  ? 

31.  A  man  has  an  acre  of  land  16  rods  in  length. 
How  wide  is  it  ? 

32.  How  many  steps  must  the  owner  take  to  walk 
round  it,  if  he  take  5  steps  to  a  rod  ? 

33.  A  man  has  an  acre  of  land  8  rods  wide.  How 
long  is  it  I  How  many  rods  of  fence  will  it  take  to 
fence  it  ? 

34.  If  a  road  4  rods  wide  is  laid  out  through  my 
land,  how  much  of  the  road  will  it  take  in  length  to 
make  an  acre  ?  How  many  acres  will  there  be  in  one 
mile  of  the  road  ? 

35.  If  it  passes  through  my  land  for  half  a  mile, 
and  I  am  paid  at  the  rate  of  30  dollars  an  acre  for  the 
land  occupied  by  the  road,  what  will  be  the  amount  o, 
damages  due  me  ? 

36.  If  land  in  the  city  is  worth  45  cents  a  square 
foot,  what  will  be  the  cost  of  a  building-lot  30  feet 
front  and  60  feet  from  front  to  rear  ? 

37.  There  are  two  pieces  of  land ;  one  of  them  12 
rods  square,  the  other  13.     Which  is  nearest  an  acre  ? 

38.  There  is  a  piece  of  land  12£  rods  square.  How 
much  does  it  fall  short  of  an  acre  ? 

39.  A  painter  tells  me  it  will  cost  20  cents  a  square 
yard  to  paint  the  floor  of  a  room  in  my  house.     Sup- 
posing the  room  is  5  yards  wide  and   6J-  yards  long, , 
what  will  the  painting  of  it  come  to? 


SQUARE    MEASURE.  105 

40.  What  will  the  painting  of  an  entry  cost,  at  the 
same  rate,  that  is  l£  yards  wide  and  7  yards  in  length  ? 

41.  A  stonecutter  agrees  to  lay  a  hammered  stone 
door-step  for  50  cents  for  every  square  foot  of  ham- 
mered surface.  The  stone  is  5  feet  long,  3|  feet  wide, 
and  9  inches  thick.  What  is  the  surface  of  the  top, 
the  two  ends,  and  the  front  edge,  added  together? 
What  will  be  the  cost  of  the  stone  ? 

42.  How  many  men  could  stand  on  |  of  a  mile 
square,  allowing  each  man  1  square  yard  to  stand  upon  ? 

There  are  various  ways  of  finding  the  answer  to 
the  above  questions.  To  encourage  the  student's  in- 
vention, some  of  them  will  be  here  suggested. 

First  Method. — As  there  are  30 \  square  yards  in  one 
square  rod,  multiply  80  (the  number  of  rods  in  one 
fourth  of  a  mile)  by  itself,  and  this  product  by  30£. 
80X80  =  6400;  6400X30*=  180,000  + 12,000+ 1600 
=  193,600,  answer. 

Second  Method.  —  Multiply  80  by  5J,  which  will 
give  the  number  of  men  in  one  line  one  fourth  of  a 
mile  long.  Multiply  this  product  by  itself.  80X5^  = 
440  ;  4402  =  160,000  +  32,000  + 1600  =  193,600,  an- 
swer. 

There  are  still  other  ways  of  solving  the  question, 
which  the  student  may  discover  for  himself. 


Note.  —  For  those  students  who  have  not  time  to  go 
through  the  book,  the  course  of  instruction  in  Mental 
Arithmetic  may  properly  close  at  this  place.  With  a 
similar  view,  the  Second  Part  may  be  divided  at  the 
close  of  Section  XXXVI.  The  ground  thus  gone 
over  tvill  be  found  to  embrace  all  the  principles  and 
practice  needed  in  the  transactions  of  ordinary  busi- 
ness. Those  iv hose  opportunities  permit  should  have 
fhe  advantage  of  the  higher  discipline  furnished  in  the 
remainder  of  the  book. 


106  MENTAL    ARITHMETIC. 


SECTION    XVI. 

CONSTRUCTION   OF   THE   SQUARE. 

If  I  place  three  dots  in  a  row,  and  place  three  such 
rows  side  by  side,  this  will  represent  to  the  eye  the 
square  of  the  number  3. 

In  the  same  way  you  may  repre- 

Thus,    •   •  •     sent  the  square  of  4,  5,  or  any  num- 

©   ©   ©     ber  whatever.     I  will  now  ask  your 

©  ©  ©     attention  to  the  square  of  4.     We 

may  make  it  by  making  a  row  of  4 

dots,  and  placing  4  such  rows  side  by  side.     But  there 

is  another  way  of  coming  at  the  square  of  4.     We 

will  take  the  square  of  3,  as  shown  above,  and  see 

what  additions  we  must  make  to  it,  in  order  to  make 

it   the   square  of  4.     You   observe    that    it    must   be 

wider  by  one  row,  and  longer  by  one  row,  than  it 

is  now.     We  will  then  add  a  row  above  the  others, 

and  also  a  row  on  the  right  hand. 

I  have  made  the  additions  by 
stars  to  distinguish  them  from 
the  dots.  You  now  see  there  is 
something  wanting  to  complete 
the  square, — a  single  star  in  the 
corner. 


Thus, 

*  *  * 

•  •   •  * 

•  •  •  * 

©   ©   ©   * 

Thus, 

*  *   *  * 

•  •  •  * 

•  •   •  * 

•   •  •   * 

You  observe,  therefore,  that 
you  obtain  the  square  of  4  by 
adding  to  the  square  of  3  twice 
3  plus  1.  We  will  now  take  the 
square  of  4,  and  by  additions  to 
it  obtain  the  square  of  5.  Adding  a  row  of  4  at  the 
top,  and  a  row  of  4  at  the  right  hand,  there  will  be 
one  wanting  at  the  corner  to  complete  the  square.' 


CONSTRUCTION    OF    THE    SOJJARE.  107 

Adding  this,  which  makes  twice  4+1,  we  have  the 
square  complete.  If,  therefore,  we  have  the  square  of 
any  number,  we  can  find  the  square  of  a  number  one 
greater  by  adding  twice  the  first  number  plus  1. 

The  square  of  5  is  25.  What  must  you  add  to  this 
square  to  make  the  square  of  6? 

What  must  you  add  to  the  square  of  6  to  make  the 
square  of  7  ?  What  must  you  add  to  the  square  of  7 
to  make  the  square  of  8  ? 

What  must  you  add  to  the  square  of  9  to  make  the 
square  of  10? 

The  square  of  15  is  225.  What  is  the  square  of  16, 
by  the  above  method  ? 

The  square  of  20  is  400.    What  is  the  square  of  21  ? 

The  square  of  30  is  900.    What  is  the  square  of  31  ? 

The  square  of  40  is  1600.  What  is  the  square  of  41  ? 

The  square  of  50  is  2500.  What  is  the  square  of  51? 

What  is  the  square  of  60  ?  Of  61  ?  Of  70  ?  Of  71  ? 
Of80?    Of 81?    Of90?    Of91? 

We  will  now  return  to  the  square  of  3 ;  and  I  ask 
your  close  attention  once  more.  Supposing  we  have 
the  square  of  3  before  us,  and  we  wish  to  make  such 
additions  to  it  as  shall  make  the  square  of  5.  As 
5  is  2  greater  than  3,  we  must  add  2  rows  instead 
of  1.  If  we  add  2  rows  of  3  at  the  top,  and  2 
rows  of  3  at  the  right  hand,  the  figure  will  stand 
thus, 

*  *    *    O    O        Here  you  see  there  are  four 

*  *    *    O    O    stars  wanting  to  complete  the 

square.     I  have  marked  their 

*  9    ©    *    *    places  by  the  circle,  O.    If  you 

*  •    •     *    *    suppose  these  four  to  be  added, 

the   square  will  be  complete, 

*  ®    *  and  will  be   the  square  of  5. 
,  The  question  is,  now,  What  has  been  added  to  the 

square  of  3  in  order  to  make  the  square  of  5?     You 


108  MENTAL    ARITHMETIC. 

observe  there  are  added  6  stars,  or  two  rows  of  3  at 
the  top,  6  on  the  right  hand,  and  4  in  the  corner,  to 
make  the  square  of  5.  But  we  can  express  this  in  a 
different  way.  We  may  consider  5  as  consisting  of 
two  parts,  3  and  2  added  together.  We  will  call  3  the 
first  part,  and  2  the  second  part,  of  5.  Now,  by  the 
figure,  you  perceive  that  the  square  of  5  is  made  up, 
first,  of  the  square  of  the  first  part,  that  is,  the  nine 
dots ;  then  the  stars  at  the  top  are  the  product  of  the 
first  part  multiplied  by  the  second;  and  adding  to 
these  the  stars  on  the  right  hand,  we  have  twice  the 
product  of  the  first  part  into  the  second;  and,  last, 
we  have,  in  the  corner,  the  square  of  the  second 
part. 

To  state  it  briefly  once  more :  Regarding  5  as 
made  up  of  the  two  parts,  3  and  2,  the  square  of  5, 
we  find,  is  equal  to  the  square  of  the  first  part  +  twice 
the  product  of  the  two  parts  +  the  square  of  the  second 
part. 

This  is  called  expressing  the  amount  of  a  square  in 
the  terms  of  its  parts. 

Examine  and  answer  the  following  questions:  — 

1.  If  we  regard  the  number  6  as  made  up  of  two 
parts,  4  and  2,  how  will  you  express  the  square  of  6 
in  the  terms  of  its  parts? 

2.  Regard  the  number  7  as  consisting  of  two  parts, 
5  and  2.  What  is  the  square  of  7  in  the  terms  of  its 
parts  ? 

You  can  draw  the  figure  for  yourself,  and  see  the 
application  of  the  principle  in  the  above  cases. 

It  is  of  no  consequence  in  what  way  the  number  is 
divided.  The  operation  will  bring  out  the  exact 
square  of  the  whole  number  in  all  cases.  To  show 
this,  we  will  take  the  number  10,  the  square  of  which 
is  100.  We  will  first  divide  10  into  the  parts  7  and 
3;  then,  by  the  formula  given  above,  the  sq.  of  7  + 
twice  the  product  of  7  into  3  +  sq.  of  3,  will  be  the* 


THE  SQUARE  OF  FRACTIONAL  NUMBERS.     109 

sq.  of  the  whole  number.  The  sq.  of  7  =  49;  twice 
7X3=42;  the  sq.  of  3  =  9;  49  +  42  +  9=100. 

We  will  now  divide  10  into  the  parts  6  and  4,  pro- 
ceeding as  above.     We  find  36  +48  + 16  =  100. 

Again;  we  will  divide  10  into  the  equal  parts  5 
and  5.     25  +  50  +  25=100. 

Finally;  divide  10  into  the  parts  8  and  2.  64  +  32 
+  4=100. 

We  will  now  apply  the  above  method  to  the  pur- 
pose of  finding  some  squares  of  larger  numbers. 

3.  What  is  the  sq.  of  25?  Dividing  into  20  +  5; 
Ans.  400  +  200  +  25  =  625. 

4.  What  is  the  sq.  of  35,  or  30  +  5?  Ans.  900  + 
300  +  25=1225. 

5.  What  is  the  sq.  of  46,  or  40  +  6?  Ans.  1600  + 
480  +  36  =  2116. 

6.  What  is  the  sq.  of  55}  Of  64?  Of  75}  Of  83? 
Of  92? 

7.  What  is  the  sq.  of  125?  Divide  into  100  +  25. 
100  sq.  =  10,000;  twice  100X25=5000;  25sq.  =  625. 
Ans.   15,625. 

8.  What  is  the  square  of  150?    Of  230?    Of  510? 

The  same  formula  will  embrace  the  examples  men- 
tioned in  the  first  part  of  this  section,  when  the  second 
part  of  the  number  is  1.     For  example, 

9.  What  is  the  sq.  of  5,  or  4  + 1  ?  Here  twice  the 
product  of  the  two  parts  is  merely  twice  the  first  part, 
inasmuch  as  multiplying  by  1  does  not  increase  the 
number;  and  the  sq.  of  1  is  only  1.  The  answer, 
therefore,  by  the  formula,  is  16  +  8  + 1  =  25. 

10.  This  method  will  apply  to  the  squaring  a  whole 
number  and  a  fraction,  as  follows :  What  is  the  sq.  of 
l+£?  Ans.  l  +  l+£  =  2£;  for  twice  the  product  of 
£  into  1  is  1,  and  the  sq.  of  £  is  £. 

To  test  the  correctness  of  this  answer,  we  will  per- 
form the  operation  another  way.   14-  =  £ ;   g  sq.  =  £  =  2£. 

11.  What  is  the  sq.  of  2}  ?    3f?"  4£?    5£?    6£?   7  p. 

10 


110  MENTAL    ARITHMETIC. 

The  answer,  in  each  of  these  cases,  may  be  tested 
by  changing  the  mixed  number  to  an  improper  frac- 
tion ;  as,  2f  =  £,  &c. 

We  may  in  the  same  way  square  the  sum  ot  two 
fractions. 

12.  What  is  the  sq.  of  £  +  £?  Ans.  i  +  i  +  i=l. 
Now,  |-|-^z=l,  the  square  of  which  is  1. 

13.  What  is  the  sq.  of  £  +  £  ?  Ans.  T%^  T\  +  TV  = 
\%  or  1.     Now,  f -f- j-=  1,  the  sq.  of  which  is  1. 

14.  What  is  the  sq.  of  f  +  £?  The  sum  of  these  is 
2,  the  sq.  of  which  is  4  ;  the  answer,  therefore,  should 
be  4.  Applying  the  formula,  the  operation  is  as  fol- 
lows;  £  +  £  +  £  =  .^.=  4. 

PRACTICAL   QUESTIONS. 

We  will  now  apply  the  above  principle  to  the  solu- 
tion of  some  questions  which  appear  at  first  a  little 
difficult. 

1.  A  boy  had  some  apples.  He  placed  part  of  them 
in  rows  making  a  square,  and  found  he  had  6  apples 
left.  He  placed  another  row  on  two  sides,  and  found 
he  had  enough  to  complete  the  square  except  one 
apple  at  the  corner.  How  many  apples  were  there  in 
the  first  square  ?    How  many  apples  had  he  ? 

2.  Three  boys  were  playing  at  marbles.  The  first 
says,  "  I  have  just  marbles  enough  to  make  a  square  ;  " 
and  he  placed  them  in  rows  on  the  floor,  forming  a 
square.  The  second  boy  says,  "  I  have  twelve  mar- 
bles, and  I  will  put  a  row  on  two  sides  of  yours,  and 
make  your  square  larger;"  but, on  placing  his  marbles, 
he  found  he  wanted  3  more  to  complete  the  square. 
Then  the  third  boy  says,  "  I  have  just  three,  and  that 
will  make  the  square  complete." 

How  many  had  the  first  boy  ?  How  large  was  the 
square  which  all  the  marbles  made  ? 

3.  The    boys   of    a  school   thought,    one   day,   at 


COMPLETING  OP  THE  DEFECTIVE  SQUARE.    Ill 

recess,  they  would  form  themselves  into  a  square.  A 
part  of  them  first  formed  a  square,  when  it  was  found 
that  there  were  15  boys  left.  These  15  then  placed 
themselves  in  a  row  on  two  sides  of  the  square,  when 
it  was  found  that  it  required  2  boys  more  to  complete 
the  square.  How  many  boys  were  there  in  the  first 
square  ?     How  many  in  all  ? 

4.  A  general,  drawing  up  his  soldiers  in  a  square 
body,  with  the  same  number  in  rank  and  file,  found 
he  had  55  men  over  and  above.  He  placed  these  in  a 
row  on  two  sides,  and  found  that  he  now  wanted  30 
men  to  complete  the  square.  How  many  men  were 
there  on  a  side  of  the  first  square  ?  How  many  men 
in  the  first  square  ?     How  many  men  had  he  in  all  ? 

5.  There  is  a  certain  square  number  expressed  in 
the  terms  of  its  parts ;  that  is,  it  is  expressed  in  three 
terms,  the  first  of  which  is  the  square  of  the  first  part, 
the  second  is  twice  the  product  of  the  two  parts,  and 
the  third  is  the  square  of  the  second  part.  Now,  the 
first  two  terms  are  16+24.  What  is  th*e  third  term? 
What  is  the  number  ? 

6.  There  is  a  square  number  expressed  in  the  terms 
of  its  parts.  The  first  two  terms  are  9  +  24.  What 
is  the  third  ? 

7.  The  first  and  last  terms  of  a  square  are  4  +  25. 
What  must  be  the  middle  term  ?    What  is  the  number  ? 

8.  The  first  and  last  terms  of  a  square  are  9  +  4. 
What  is  the  second  term  ?     What  is  the  square  ? 

9.  Complete  the  square  whose  first  two  terms  are 
16  +  40  +  D. 

10.  Complete  the  sq.  36 +24+ a. 

11.  Complete  the  sq.  4+4+ □. 

12.  Complete  the  sq.  9  +  6  +  a. 

13.  Complete  the  sq.  16  +  8  +  d 

14.  Complete  the  sq.  25  +  10  +  Q 

15.  Complete  the  sq.  25 +  20  + a. 

16.  Complete  the  sq.  36 +  72  + a- 


1  12  MENTAL    ARITHMETIC. 

17.  Complete  the  sq.  25  +  30  +  L3 

.18.  Complete  the  sq.  25  +  40  +  □• 

The  square  root  is  the  number  which,  multiplied 
into  itself,  produces  the  square.  Thus  3  is  the  sq.  root 
of  9,  2  is  the  sq.  root  of  4,  5  is  the  sq.  root  of  25. 

The  square  root  of  a  square  of  three  terms,  like 
those  given  above,  is  the  square  root  of  the  first  term, 
plus  the  square  root  of  the  third ;  for  these,  multiplied 
by  themselves,  will  produce  the  square.  Thus  the 
square  root  of  the  square  9+12  +  4,  is  3  +  2,  or  5. 
3  is  the  sq.  root  of  9,  and  2  the  sq.  root  of  4.  This 
number,  3  +  2,  multiplied  by  itself,  will  produce  the 
square  of  9+12  +  4. 

19.  Complete  the  sq.  36  +  60  +  d.  What  is  its 
root? 

20.  Complete  the  sq.  36+12  +  CZI.  What  is  its 
root? 

21.  Complete  the  sq.  16  +  4  +  a.    What  is  its  root? 

22.  Complete  the  sq.  9  +  12  +  d    What  is  its  root? 

23.  What  is  the  sq.  root  of  169? 

Divide  the  number  into  three  terms,  100  +  60  +  9. 
We  divide  it  so,  because  100  is  a  square,  and  60  is 
twice  the  product  of  10,  the  root  of  the  1st  term,  into 
the  root  of  what  this  way  of  dividing  leaves  us  for  the 
3d  term.  That  is,  if  we  take  60  for  the  2d  term,  we 
leave  9  for  the  3d  term,  and  this  is  as  it  should  be,  for 
60  is  twice  the  product  of  10  into  3. 

The  root  is  therefore  10  +  3,  or  13. 

24.  What  is  the  sq.  root  of  196  ? 

We  will  take  for  the  1st  term  100,  whose  root  is 
10.  Now,  as  the  2d  term  is  twice  the  product  of  the, 
two  terms  of  the  root,  if  we  divide  half  of  it  by  the 
1st,  it  will  give  the  2d  term  of  the  root ;  or,  what  is  the 
same  thing,  if  we  divide  the  2d  term  of  the  square  by 
twice  the  1st  term  of  the  root,  it  will  give  the  2d  of 
the  root.  Now,  96  contains  the  2d  and  3d  terms  of 
the  square.     We  must  separate  it  into  two  parts,  such 


COMPLETING  OF  THE  DEFECTIVE  SQUARE.    113 

that  the  first  part,  divided  by  twice   10,  or  20,  will 
give  for  quotient  the  root  of  the  second  part. 

Let  us  first  try  by  dividing  it  into  60  and  36.  Now, 
60  divided  by  20  gives  3,  which  is  not  the  root  of  36. 
.  Our  division,  therefore,  was  wrong.  The  2d  term  was 
too  small,  and  the  3d  too  great.  We  will  try  again. 
By  dividing  it  into  80  and  16,  we  find  that  80  divided 
by  20  gives  4,  which  is  the  root  of  16.  The  number 
196,  when  arranged  in  the  three  terms  of  the  square, 
will  be  100  +  80+16,  and  the  root  is  10  +  4,  or  14. 

25.  What  is  the  root  of  225  ?  Here  we  must  not 
take  for  our  1st  term  200,  for  this  is  not  a  square. 
We  must  take  the  largest  square  whose  root  is  an  even 
10.  This  is  100.  We  have  remaining  125.  This 
we  must  divide  into  two  terms,  such  that  the  1st  di- 
vided by  twice  10  will  give  the  root  of  the  2d  term. 
We  will  first  divide  into  80  and  45.  80  divided  by 
20  gives  4,  which  is  not  the  square  root  of  45.  We 
will  divide  into  100  and  25.  100  divided  by  20 
gives  5,  which  is  the  exact  root  of  25.  The  number 
225,  therefore,  when  arranged  in  the  three  terms  of  a 
square,  stands  100+100  +  25,  and  its  square  root  is 
10+5,  or  15. 

26.  What  is  the  square  root  of  256  ?  Taking  for  the 
1st  term  100,  it  remains  to  divide  the  remainder,  156, 
according  to  the  principle  stated  above.  Now,  120 
will  contain  20,  6  times,  which  is  the  root  of  the  re- 
mainder, 36.  The  number  stands,  therefore,  100  + 
120  +  36;  square  root,  10  +  6,  or  16. 

27.  What  is  the  square  root  of  484?  Here  we. 
take  for  the  1st  term  400,  for  that  is  the  largest  square 
whose  root  is  in  even  tens;  its  root  is  20.  The  re- 
mainder we  may  divide  into  80  and  4.  Dividing  80 
by  twice  20,  or  40,  we  have  for  the  quotient  2,  which 
is  the  root  of  the  3d  term.  The  square,  therefore,  is 
,400  +  80  +  4;  the  root,  20  +  2,  or  22. 

10*  H 


114  MENTAL    ARITHMETIC. 

28.  What  is  the  square  root  of  529  ?  Of  576  ?  Of 
625?    Of  676? 

29.  The  following  is  a  ready  method  of  squaring  a 
mixed  number  whose  fraction  is  £;  as,  2£,  3£,  4£. 
The  fraction  will  be  £.  For  the  other  number,  mul- 
tiply the  whole  number  by  a  number  greater  than 
itself  by  1.  Thus  the  square  of  2£;  the  fraction  is 
i,  the  whole  number  2X3,  or  6;  6£.  The  square  of 
3^ ;  3X4,  or  12,  and  £.  The  sq.  of  4^  is  4X5,  or  20, 
and  {.  What  is  the  sq.  of  5±1  6£?  7£  ?  8£?  9£? 
10i? 

The  same  principle  will  apply  to  the  square  of  whole 
numbers  whose  last  figure  is  5 ;  as,  25,  45,  55,  &c. ; 
for  such  a  number  consists  of  a  certain  number  of 
tens,  and  half  of  10.  As  the  right-hand  figure  is  5, 
the  two  right-hand  figures  of  the  square  must  be  25. 
Then  multiply  the  number  at  the  left  of  the  5  by  itself 
increased  by  1 ;  and  this,  read  at  the  left  hand  of  the 
25,  will  be  the  square.  Thus,  for  the  square  of  25, 
the  two  right-hand  figures  will  be  25 ;  for  the  rest, 
multiply  2  by  3,  which  is  6.     Ans.  625. 

30.  What  is  the  square  of  35?  3X4  =  12.  To  this 
annex  25.     1225,  Ans. 

What  is  the  square  of  45?  Of  551  Of  65?  Of 
751    Of  85?    Of  95? 


SECTION   XVII. 

PRACTICAL  QUESTIONS  IN  SQUARE  MEASURE. 

1.  How  many  square  rods  are  there  in  a  square 
mile  ? 

2.  How  many  acres  are  there  in  a  square  mile?  . 


MENSURATION.  115 

3.  Divide  a  square  mile  into  4  equal  farms.  How 
many  acres  would  there  be  in  each? 

4.  How  many  acres  are  there  in  one  fourth  of  a 
mile  square  ? 

5.  How  many  acres  are  there  in  a  town  6  miles  long 
and  5  miles  broad  ? 

6.  If  half  of  the  town  is  unfit  for  improvement,  in 
consequence  of  water  and  mountains,  how  many  farms 
of  100  acres  might  be  made  from  the  other  half? 

7.  A  man  bought  a  rectangular  piece  of  land  con- 
taining 40  acres.  On  going  out  with  his  son  to  meas- 
ure round  it,  to  ascertain  how  much  fence  it  would 
require  to  enclose  it,  they  found  the  first  side  they 
measured  to  be  160  rods.  "  We  need  not  measure 
any  more,"  said  the  son,  "  for  I  can  tell  all  the  rest  in 
my  head."  How  wide  was  the  piece  ?  and  how  many 
rods  of  fence  would  it  take  to  go  round  it  ? 

8.  A  man  bought  7  acres  of  land,  in  rectangular 
form.  The  width  of  it  was  28  rods.  What  was  its 
length  ? 

If  a  four-sided  piece  of  land  is  rectangular,  its  con- 
tents may  be  found  by  multiplying  two  adjacent  sides, 
or  sides  that  meet  and  form  a  corner. 

Thus,  if  a  piece  is  12  rods  long  and  4  rods   wide, 
12  c    the    two    boundaries,    12   and   4, 

which  meet  and  form    the   right 
4  angle  at  c,  will,  when  multiplied 
together,    give     the    contents    in 
square  rods  ;   12X4  =  48. 

If  the  opposite  sides  are  parallel,  but  the  angles  are 
not  right  angles,  the  distance  between  the  two  sides 
must  be  measured  by  a  perpendicular  line,  thus : 

the  length,  16  rods,  multi- 


plied  by  the  width,  4  rods, 
will  give  the  contents. 


If  the  piece  is  a  triangle,  there  must  be  a  perpen- 


116  MENTAL     ARITHMETIC. 

•dicular  line  drawn  to  the  longest  side  from  the  an- 
gle opposite  to  it.  This  perpendicular  we  may  call 
the  height  of  the  triangle,  and  the  longest  side,  its 
length ;  and  the  height  multiplied  into  the  length, 
will  give  double  the  area ;  dividing  this  by  2,  we  get 
the  area. 

To  show  the  reason   of  this,  take  the  following 

iigure.    By  examining  this,  you  will  see  that  there  are 

16  in  it  two  triangles  just  alike. 

/p=:=^ZI~  -J  The  length   of  each   is   16 

/    t  -— ^^/     rods,  and  the  height  4  rods. 

Now,  16X4  will  give,  as  in 
the  case  above,  the  area  of  the  whole  figure,  that  is, 
■of  both  the  triangles  ;  therefore  it  will  give  twice  the 
area  of  one  Of  them. 

9.  What  is  the  area  of  a  triangle  whose  longest  side 
is  16  rods,  and  the  perpendicular,  from  the  opposite 
angle  to  this  side,  12  rods  ? 

10.  What  is  the  area  of  a  triangle  whose  longest 
side  is  24  rods,  and  its  height  9  rods  ? 

11.  What  is  the  area  of  a  triangle  whose  longest  side 
is  18  rods,  and  height  16  rods  ? 

12.  A  triangle  contains  just  one  acre.  Its  longest 
side  is  20  rods.  How  long  must  the  perpendicular  be 
from  the  opposite  angle  to  that  side  ? 

13.  A  triangle  contains  2  acres.  Its  longest  side  is 
32  rods.  How  long  is  the  perpendicular,  from  the  op- 
posite angle  to  this  side  ? 

In  a  right-angled  triangle,  the  longest  side  is  called 
the  hypotenuse ;  the  sides  containing  the  right  angle 
are  called  the  legs,  or  one  the  base  and  the  other  the 
perpendicular.  In  all  right-angled  triangles,  the  square 
of  the  hypotenuse  is  just  equal  to  the  sum  of  the 
squares  of  the  two  other  sides.  This  important 
principle  is  exhibited  to  the  eye  in  the.  following 
figure :  — 


MENSURATION. 


117 


The  hypotenuse  is 
divided  into  5  equal 
parts,  and  its  square 
is  therefore  25.  The 
base  has  4  equal  parts 
of  the  same  length, 
making  a  square  of 
16.  The  perpendic- 
ular is  divided  into 
3  equal  parts,  of  the 
same  length  as  the 
others,  which  makes 
a  square  of  9.  The 
square  of  the  perpen- 
dicular and  of  the 
base,  added  together, 
16  +  9  =  25,  which  is  the  square  of  the  hypotenuse. 

If  we  know  the  square  of  the  hypotenuse,  we  know 
the  sum  of  the  squares  of  the  two  legs.  If  we  know 
the  sum  of  the  squares  of  the  two  legs,  we  know  the 
square  of  the  hypotenuse.  If  we  know  the  square  of 
the  hypotenuse  and  of  one  leg,  we  can  find  the  square 
of  the  other  leg.  And  if  we  know  the  square  of  any 
one  of  these  sides,  we  can  obtain  the  length  of  the 
side  by  extracting  the  square  root. 

14.  In  a  certain  right-angled  triangle,  the  square  of 
the  hypotenuse  is  100  feet.  What  is  the  length  of  the 
hypotenuse?  —  In  the  same  triangle,  the  square  of  the 
base  is  64  feet.     What  is  the  length  of  the  base  ? 

In  the  same  triangle,  what  must  be  the  square  of 
the  perpendicular  ?  What  is  the  length  of  the  perpen- 
dicular? 

15.  A  and  B  set  out  from  the  same  place.  A  trav- 
els east  6  miles.  B  travels  north  till  he  is  10  miles  in  a 
straight  line  from  A.  How  far  north  has  B  travelled  ? 
>  16.  There  is  a  triangle,  the  perpendicular  of  which 
is  3  feet,  the  hypotenuse  is  5  feet.  How  long  is  the 
base? 


118 


MENTAL    ARITHMETIC. 


17.  A  man  had  a  piece  of  land,  in  the  form  of  a 
right-angled  triangle,  the  two  legs  of  which  were  equal 
to  each  other,  and  the  square  of  the  hypotenuse  was 
128  rods.     How  many  rods  were  there  in  the  piece  ? 


The  circumference  of  a 
circle  is  3  times  and  \  greater 
than  the  diameter.  If  the 
diameter  is  1  foot,  the  circum- 
ference will  be  3f  feet ;  if  the 
diameter  is  2  feet,  the  circum- 
ference will  be  6f  feet. 


18.  If  the  diameter  of  a  circle  is  3  feet,  what  will 
be  the  circumference  ?  If  the  diameter  is  4  feet  ?  If 
5  feet  ?     If  6  feet  ?     If  7  feet  ? 

If  the  diameter  of  a  water-wheel  is  16  feet,  what  is 
the  circumference  ? 

19.  If  the  diameter  of  the  earth  is  8000  miles,  what 
is  the  circumference  ? 

To  find  the  area  of  a  sector  of  a  circle,  as  a,  e,  b, 
multiply  the  arc  by  half  the  radius.  This  figure  may 
be  regarded  as  a  triangle,  the  base  of  which  is  the  arc, 
and  the  radius  the  height ;  and  you  have  seen  before, 
that,  in  a  triangle,  the  base  multiplied  by  half  the 
height  gives  the  area.  From  this  we  may  obtain  a 
method  of  obtaining  the  area  of  the  whole  circle. 

Multiply  the  circumference  by  half  the  radius.  For 
we  may  regard  the  circle  as  made  up  of  a  great  num- 
ber of  small  triangles,  whose  bases  added  together  are 
the  circumference  of  the  circle,  and  whose  height  is 
equal  to  radius,  being,  in  each  case,  the  distance  from 
the  circumference  to  the  centre. 

20.  What  is  the  circumference  of  a  circle  whose t 
diameter  is  14  feet  ? 


ANALYSIS    OF    PROBLEMS.  119 

What  is  its  area  ? 

21.  What  is  the  circumference  of  a  circle  whose 
diameter  is  12  feet. 

What  is  its  area  ? 

22.  What  is  the  circumference  of  a  circle  whose 
diameter  is  20  feet  ? 

What  is  its  area  ? 

23.  What  is  the  circumference  of  a  circle  whose 
diameter  is  28  feet  ? 

What  is  the  area? 


SECTION    XVIII. 

ANALYSIS   OF   PROBLEMS. 

1.  A  boy  spent  one  half  the  money  he  had,  and  had 
1  dollar  left.     How  much  had  he  at  first  ? 

2.  A  boy  spent  one  third  of  the  money  he  had,  and 
had  1  dollar  left.     How  much  had  he  at  first  ? 

Ans.  If  he  lost  one  third,  he  had  two  thirds  left.  If 
1  dollar  was  two  thirds,  half  a  dollar  must  be  one 
third,  and  f  of  a  dollar  the  whole.  He  had  1  dollar 
and  a  half. 

3.  A  boy  spent  J  of  his  money,  and  had  1  dollar 
left.     How  much  had  he  at  first  ? 

Let  this  and  the  following  answers  be  given  in  form 
of  a  fraction,  like  the  preceding  answer. 

4.  A  boy  spent  |  of  his  money,  and  had  1  dollar 
left.     How  much  had  he  at  first  ? 

5.  A  boy  spent  £  of  his  money,  and  had  1  dollar 
left.     How  much  had  he  at  first  ? 

6.  A  boy  spent  }  of  his  money,  and  had  1  dollar 
left.     How  much  had  he  at  first  ? 

7.  A  boy  spent  £  of  his  money,  and  had  1  dollar 
left.     How  much  had  he  at  first  ? 


120  MENTAL    ARITHMETIC. 

8.  A  boy  spent  £  of  his  money,  and  had  1  dollar 
left  ?     How  much  had  he  at  first  ? 

9.  A  boy  spent  TV  of  his  money,  and  had  1  dollar 
left.     How  much  had  he  at  first? 

10.  A  man  carried  some  corn  to  mill.  The  miller 
took  TV  of  it  for  toll,  and  then  there  was  just  a  bushel. 
How  much  did  the  man  carry  to  mill  ? 

11.  A  man  carried  some  cloth  to  be  fulled.  It 
shrank  two  sevenths  in  its  length,  and  was  then  just  a 
yard  long.     How  long  was  it  at  first  ? 

12.  A  man  drew  a  prize  in  a  lottery.  ?V  °f  the  prize 
was  retained,  and  then  the  drawer  received  just  100 
dollars.     How  much  was  the  prize  ? 

13.  If  a  stick  of  timber  shrink  T\  in  weight  in  sea- 
soning, and  then  weigh  100  pounds,  how  much  did  it 
weigh  at  first  ? 

14.  A  teamster  sold  f-  of  a  cord  of  wood,  and  then 
had  half  a  cord  left.     How  much  had  he  at  first  ? 

15.  A  man  had  an  estate  left  him  by  his  father.  He 
lost  |  of  it.  He  then  received  1000  dollars,  and  then 
he  had  3500  dollars.     How  much  had  he  at  first  ? 

16.  A  merchant  began  trade  with  a  sum  of  money, 
and  gained  so  as  to  increase  his  original  stock  by  I  of 
itself.  He  then  lost  500  dollars,  and  had  4500  dollars 
left.     How  much  did  he  begin  with? 

17.  A  man  set  out  on  a  journey,  and  spent  half  the 
money  he  had  for  a  dinner.  He  then  paid  half  of  what 
he  had  left  for  provender  for  his  horse  ;  then,  half  of 
what  now  remained  for  toll  in  crossing  a  bridge ;  and 
had  10  cents  left.     How  much  had  he  at  first  ? 

18.  A  boy  spent  §  of  his  money  for  a  book,  and  f  of 
it  for  some  paper,  and  had  8  cents  left.  How  much 
had  he  at  first  ? 

19.  A  boy,  playing  at  marbles,  lost,  in  the  first  game, 
I  of  what  he  had  ;  in  the  second  game,  i  of  what  he 
then  had ;  in  the  third,  -\  of  what  he  then  had ;  in  the 
fourth,  11;  and  then  he  had  16  marbles  left.  How 
many  had  he  at  first? 


ANALYSIS  OF  PROBLEMS.  121 

20.  A  boy,  playing  at  marbles,  wins,  in  the  first 
game,  so  as  to  double  the  number  of  marbles  he  had  ; 
in  the  second  game,  he  loses  £  of  what  he  then  had  ; 
in  the  third  game,  he  loses  5,  and  then  finds  he  has 
just  as  many  as  at  first.     How  many  had  he  at  first  ? 

21.  A  man  had  his  sheep  in  three  pens.  In  the 
first,  there  were  10  sheep ;  in  the  second,  there  were 
as  many  as  in  the  first,  and  half  the  number  in  the 
third ;  in  the  third,  there  were  as  many  as  in  the  first 
and  second.     How  many  had  he  in  all  ? 

22.  In  an  orchard,  i  of  the  trees  are  plum-trees; 
there  are  20  cherry-trees ;  and  the  apple-trees,  which 
constitute  the  remainder,  are  half  as  many  as  the  plum 
and  cherry-trees  added  together.  How  many  trees 
are  there  in  the  orchard  ? 

23.  John  and  William  were  talking  of  their  ages. 
John  says,  "lam  twelve  years  old."  William  says, 
"  If  half  my  age  were  multiplied  by  one  fourth  of 
yours,  and  half  your  age  plus  one  subtracted  from  the 
product,  that  would  give  my  age."     How  old  was  he  ? 

24.  A  man,  talking  of  the  age  of  his  two  children, 
said  the  youngest  was  three  years  old ;  the  age  of  the 
eldest  was  £  his  own  age  ;  if  his  own  age  was  divided 
by  that  of  his  youngest,  and  once  and  one  third  the 
age  of  the  youngest  subtracted  from  the  quotient,  that 
would  give  the  age  of  the  eldest.  How  old  was  the 
eldest  ? 

25.  The  number  of  pupils  in  a  school  is  such  that, 
if  you  take  half  of  them,  and  increase  that  by  2 ;  then 
take  one  third  of  this  last  number,  and  increase  it  by 
3;  and  from  this  number  subtract  6;  the  remainder 
will  be  7.     How  many  are  there  in  the  school  ? 

26.  A  boy  plays  three  games  at  marbles.  In  the  first, 
he  loses  a  certain  number ;  in  the  second,  he  gains  8 ; 
in  the  third,  he  loses  4 ;  and  then  he  finds  he  has  2 

j  more  than  he  began  with.     How  many  did  he  lose  in 
the  first  game  ? 
11 


122  MENTAL    ARITHMETIC. 

27.  A  boy,  playing  at  marbles,  first  lost  one  third 
of  what  he  had ;  he  then  doubled  his  number,  when 
he  had  5  marbles  more  than  he  had  at  first.  How 
many  had  he  at  first  ? 

28.  There  is  a  certain  number,  one  third  of  which 
exceeds  one  fourth  of  it  by  2.     What  is  the  number  ? 

29.  There  is  a  certain  number,  one  fourth  of  which 
exceeds  one  fifth  of  it  by  1.     What  is  the  number  ? 

30.  There  is  a  certain  number,  one  third  of  which 
added  to  one  fifth  of  it  amounts  to  16.  What  is  the 
number  ? 

31.  There  is  a  number,  one  third,  one  fourth,  and 
one  fifth  of  which,  added  together,  are  94.  What  is 
the  number  ? 

32.  What  is  that  number,  a  fifth  of  which  exceeds 
a  sixth  of  it  by  4  ? 

33.  What  number  is  that,  of  which  a  fourth  part 
exceeds  a  seventh  part  by  9  ? 

34.  In  a  certain  orchard  there  are  apple,  peach,  and 
pear-trees.  The  apple-trees  are  2  more  than  half  the 
whole.  The  peach-trees  are  one  third  of  the  whole, 
and  are  14  less  than  the  apple-trees.  The  rest  are 
pear-trees.  How  many  are  there  of  each  kind  ?  and 
how  many  in  all  ? 


SECTION   XIX. 

SOLID  MEASURE. 

Whatever  has  length,  and  breadth,  and  thickness,  is 
a  solid.  A  block  of  wood  1  inch  long,  1  inch  high, 
and  1  inch  wide,  is  a  solid  inch.  A  block  1  foot  long, 
1  foot  wide,  and  1  foot  high,  is  a  solid  foot.  A  block. 
1  yard  long,  1  yard  wide,  1  yard  high,  is  a  solid  yard. 


SOLID    MEASURE.  123 

1.  How  many  solid  inches  are  there  in  a  block  3  in. 
long,  2  in.  wide,  and  1  in.  high  ? 

2.  How  many  in  a  block  4  in.  long,  3  in.  wide,  and 
2  m.  high  ? 

3.  How  many  solid  feet  are  there  in  a  block  5  feet 
long,  3  feet  wide,  and  2  feet  high  ? 

4.  How  many  solid  feet  in  a  block  7  feet  long,  2  feet 
wide,  and  2  feet  high  ? 

When  a  solid  has  its  length,  height,  and  breadth, 
equal  to  each  other,  it  is  called  a  cube  ;  and  the  linear 
measure  of  its  length,  height,  or  breadth,  is  called  the 
root  of  the  cube.  We  have  seen  what  is  a  cubic  inch, 
a  cubic  foot,  and  a  cubic  yard. 

Suppose,  now,  we  have  a  pile  of  cubic  inch  blocks, 
and  we  wish  to  construct  from  them  a  cube,  each  of 
whose  dimensions  shall  be  2  inches.  We  will  first 
take  2  blocks,  and  place  them  down  side  by  side. 
This  will  be  as  long  as  the  required  figure,  but  it  will 
not  be  wide  enough  nor  high  enough.  To  make  it 
wide  enough,  we  will  place  2  more  blocks  down  by 
the  side  of  the  former.  The  figure  now  contains  4 
cubic  inches,  and  is  2  inches  long  and  2  inches  wide, 
but  it  is  only  1  inch  high.  To  make  it  2  inches  high, 
we  must  place  upon  this  another  layer  of  4  blocks, 
arranged  just  like  the  former.  The  figure  will  then 
be  2  in.  long,  2  in.  wide,  and  2  in.  high  :  it  contains 
8  cubic  inches,  and  is  the  cube  of  2. 

5.  How  many  blocks  will  you  require,  and  how  will 
you  arrange  them,  to  make  the  cube  of  3  ? 

6.  How  many  blocks  will  you  require,  and  how  will 
you  arrange  them,  to  make  the  cube  of  4  ? 

7.  How  many  blocks  will  you  require,  and  how  will 
you  arrange  them,  to  make  the  cube  of  5? 

The  cube,  when  expressed  in  numbers,  is  the  same 
as  the  3d  power  of  the  root.  It  is  found  by  taking  the 
root  3  times  as  a  factor.     Thus,  the  3d  power  of  2  is 


124 


MENTAL    ARITHMETIC. 


2X2X2  =  8.     The  3d  power  of  3  is  3X3X3  =  27; 
of  4,  is  4X4X4  =  64;  of  5,  is  5X5X5=125. 

In  this  way  we  may  find  the  3d  power  of  any 
number. 

8.  How  many  blocks,  of  a  cubic  foot  each,  will  it 
take  to  form  a  cubic  solid  6  feet  on  a  side  ? 

9.  How  many  blocks,  of  a  cubic  foot  each,  will  it 
take  to  form  a  cubic  solid  of  7  feet  each  way  ? 

10.  How  many  cubic  feet  will  it  take  to  form  a 
cube  of  8  feet  ? 

11.  How  many  cubic  feet  will  it  take  to  form  a 
cube  of  9  feet  ? 

12.  How  many  cubic  feet  will  it  take  to  form  a 
cube  of  10  feet? 

13.  How  many  cubic  inches  are  there  in  a  cubic 
foot? 

14.  A  pile  of  wood  8  feet  long,  4  feet  high,  and  4 
feet  wide,  makes  a  cord.  How  many  cubic  feet  are 
there  in  a  cord  ? 

15.  A  pile  of  wood  4  feet  long,  4  feet  high,  and  1 
foot  thick,  makes  what  is  called  a  cord  foot.  How 
many  cubic  feet  are  there  in  a  cord  foot  ? 

16.  How  many  cord  feet  are  there  in  a  cord  ? 

17.  There  is  a  pile  of  wood  40  feet  long,  4  feet 
wide,  and  5  feet  high.  How  many  cords  does  it 
contain? 

18.  There  is  a  stick  of  hewn  timber  25  feet  long, 
1  foot  wide,  and  1  foot  thick.  How  many  cubic  feet 
does  it  contain  ? 

19.  There  is  a  tree,  from  the  but-end  of  which  a 
stick  may  be  hewn  13  feet  long,  2  feet  wide,  and  2 
feet  thick.     How  many  cubic  feet  will  it  contain  ? 

20.  It  is  estimated  that  50  feet  of  hewn  timber 
weigh  a  ton.  If  50  cubic  feet  weigh  20  cwt.  net 
weight,  what  will  1  foot  weigh  ? 

21.  If  you  divide  a  cubic  inch  into  blocks  meas-  t 


CONSTRUCTION    OF    THE    CUBE.  125 

uring  i  an  inch  each  way,  how  many  such  will  there 
be  in  a  cubic  inch  ? 

22.  How  many  cubic  half  inches  are  in  a  cubic 
inch  ? 

23.  If  you  divide  a  cubic  inch  into  cubes  of  i  of 
an  inch  each,  how  many  such  will  there  be  ? 

24.  How  many  cubic  quarter  inches  are  there  in  a 
cubic  inch? 

25.  How  many  cubic  inches  are  there  in  a  cube  of 
one  inch  and  a  half  ? 

26.  If  a  man  digs  a  cellar  at  the  rate  of  f  of  a  dol- 
lar for  a  cubic  yard,  what  will  the  job  come  to,  if  the 
cellar  is  18  feet  long,  12  feet  wide,  and  6  feet  deep? 

27.  A  stone-layer  agreed  to  build  a  solid  wall  30 
feet  long,  44  feet  thick,  and  6  feet  high,  for  2£  dollars 
a  cubic  yard.     What  did  the  wall  cost  ? 


CONSTRUCTION   OF  THE   CUBE. 

We  have  seen  that  the  third  power,  or  cube,  of  any 
number,  is  obtained  by  taking  the  number  three  times 
as  a  factor.     The  product  is  the  cube,  or  third  power. 

In  this  way  the  cube  of  any  number  whatever  may 
be  obtained.  There  is  another  way,  however,  of  con- 
structing the  cube,  the  knowledge  of  which  is  very  im- 
portant in  the  operation  of  extracting  the  cube  root. 

Suppose  we  wish  to  find  the  cube  of  5.  Instead  of 
taking  5  three  times  as  a  factor,  thus,  5X5X5  =  125, 
we  will  regard  the  number  5  as  consisting  of  two 
parts,  3  and  2.  We  will  call  3  the  first  part,  and  2 
the  second  part,  of  5. 

We  will  begin  by  making  the  cube  of  the  first  part, 
3,  thus,  3X3X3  =  27. 

We  will  regard  this  as  a  cube  of  3  inches,  (that  is, 
»3  in.  long,  3  in.  wide,  and  3  in.  high,)  and  represent 
it  bv  the  following  figure  :  — 
11* 


126 


MENTAL    ARITHMETIC. 


The  question  now  is,  How  shall 
we  enlarge  this  cube  of  3,  so  as 
to  make  it  the  cube  of  5  ?  It  is 
evident,  it  must  be  2  in.  longer,  2 
in.  broader,  and  2  in.  higher,  than 
it  now  is.  We  will  begin,  then, 
by  putting  2  layers  of  inch  blocks 
on  the  front  side,  2  layers  on  the 
right  side,  and  2  layers  on  the  top.  The  figure,  thus 
enlarged,  is  not  a  cube.  There  are  several  places  not 
filled  up.  It  is  nearer  the  cube  of  5  than  it  was 
before  ;  but  something  more  must  be  added.  Before 
making  that  addition,  however,  let  us  see  what  we 
have  done.  The  figure  is  the  cube  of  3,  which  is  the 
first  part  of  5.  To  this  there  are  3  equal  additions 
made.  Each  of  these  additions  is  3  in.  square,  and 
2  in.  thick.  Now,  3  is  the  first  part  of  5.  Each  ad- 
dition, therefore,  contains  the  square  of  the  first  part, 
3,  multiplied  by  the  second  part,  2  or  32X2.  There- 
fore the  three  additions  will  be  3  times  the  square  of 
the  first  part  multiplied  by  the  second. 

The  whole  figure,  therefore,  after  these  three  addi- 
tions are  made,  contains  33  +  3  times  32X2. 

We  will  now  see  what  additions  must  next  be  made 
to  the  figure. 

There  are  3  places  that  need  filling  up,  each  3  in. 
long,  2  in.  wide,  and  2  in.  high.  Each  of  these  new 
additions  is  2  in.  square  and  3  in.  long.  It  consists, 
therefore,  of  the  first  part  of  5  multiplied  by  the  square 
of  the  second ;  and  the  three  together  are  3  times  the 
first  part  multiplied  by  the  square  of  the  second. 
There  is  one  addition  wanting  to  complete  the  cube  ; 
that  is  at  the  corner.  It  must  be  2  in.  long,  2  in. 
wide,  and  2  in.  high;  that  is,  the  cube  of  2;  or, 
in  other  words,  the  cube  of  the  second  part. 

Remembering  that  the  two  parts  of  5,  as  here  di- 
vided, are  3  and  2;  the  printed  figure  is  the  cube  ot 


CONSTRUCTION    UF    THE    CUBE.  127 

the  first  part ;  the  first  addition  is  3  times  the  square 
of  the  first  part,  multiplied  by  the  second ;  the  second 
addition  is  3  times  the  first  part,  multiplied  by  the 
square  of.  the  second ;  the  third  addition  is  the  cube 
of  the  second  part.  Let  the  letter  a  stand  for  the  first 
part,  3 ;  and  the  letter  b,  for  the  second  part,  2.  The 
printed  figure  will  then  be  a3 ;  the  first  addition,  3a2b; 
the  second  addition,  3a  b2  ;  the  third  addition,  bs.  The 
whole  cube,  therefore,  will  be  a*  +  3a2b-\-3ab2 +  b3. 
Observe  that  the  letters  and  numbers  are  to  be  mul- 
tiplied together,  though  there  is  no  sign  of  multiplica- 
tion Between  them;  as,  3a2b  is  three  times  the  square 
of  a  multiplied  by  b. 

These  are  called  the  four  terms  of  the  cube,  when 
the  root  is  in  two  parts. 

If  we  express  the  above  in  the  numbers  for  the 
cube  of  5,  it  will  stand  thus; 

1st.  2d.  3d.  4th. 

^+3X32X2  +  3X3X22+?3. 

1.  What  number  makes  the  first  term  of  this  cube  ? 

2.  What  number  forms  the  second  term? 

3.  What  number  forms  the  third  term  ? 

4.  What  number  forms  the  fourth  term? 

5.  What  do  all  the  four  terms  amount  to  ? 

6.  Which  of  the  four  terms  contains  the  third  power 
of  the  first  part?  Which  contains  the  second  power 
of  the  first  part? 

7.  Which  contains  the  first  power  of  the  first  part  ? 
Which  term  contains  the  first  power  of  the  second 
part?    Which  the  second  power?    Which  the  third? 

8.  If  the  fourth  term  of  the  above  cube  were  not 
given,  how  could  you  determine  from  the  others 
what  it  must  be?  fefejfe* 

9.  If  the  third  term  were  gone,  how  could  you  restore 
it?    If  the  second  was  gone,  how  could  you  restore  it  ? 

10.  If  you  divide  the  number  5  into  the  two  parts 


128  MENTAL    ARITHMETIC. 

4  and  1,  and  express  the  cube  according  to  the  above 
rule,  what  will  the  first  term  be  ?  What  will  be  the 
second  term?  What  will  be  the  third  term?  What 
the  fourth? 

Remember,  here,  that  all  powers  of  1  are  1,  —  neither 
more  nor  less. 

Divide  the  number  6  into  4  +  2,  and  form  the  cube 
according  to  the  above  rule. 

11.  What  will  the  first  term  be?  The  second? 
The  third?     The  fourth? 

12.  What  will  they  all  amount  to  ? 

Multiply  6  into  itself  3  times,  thus,  6X6X6,  and 
see  if  it  amounts  to  the  same. 

13.  Divide  6  into  the  parts  5  +  1,  and  form  the 
cube.  What  will  be  the  first  term?  The  second? 
The  third  ?    The  fourth  ?    What  do  they  all  amount  to  ? 

14.  There  is  a  cube  in  four  terms,  the  first  two 
terms  of  which  are  33  +  3X32Xl.  What  must  be 
the  third  term?  What  the  fourth  term?  What  is  the 
number  of  the  cube  ?  What  is  the  root  of  the  cube  ? 
This  root  is  the  cube  root  of  the  first  term,  added  to 
the  cube  root  of  the  last. 

15.  There  is  a  cube  in  four  terms,  the  first  two  of 
which  are  23  +  3X22X2.  What  is  the  third  term? 
What  the  fourth?  What  is  the  whole  cube?  What  is 
the  first  part  of  the  root?  What  is  the  second  part? 
What  is  the  whole  root? 

16.  Complete  the  cube  43+3x42X2  +  C3+23. 
What  is  the  number  of  the  cube?    What  the  root? 

17.  Complete  the  cube  33  +  3X32X3  +  3X3X 
32  +  t=3. 

What  is  the  number  of  the  cube? 

18.  There  is  a  cube  in  four  terms,  the  first  of  which 
is  1000.     What  is  the  first  part  of  the  root  ? 

19.  The  second  term  of  the  same  cube  is  300X6, 
or  1800.     What  is  the  third  term? 

20.  What  is  the  fourth  term  of  the  same  cube  ? 


RATIO. PROPORTION.  129 

21.  What  is  the  root  of  the  above  cube? 

22.  The  first  term  of  a  cube  is  1000  j  the  second  is 
300X8,  or  2400.     What  is  the  third  term? 

23.  What  is  the  fourth  term  of  the  above  cube? 

24.  The  first  term  of  a  cube  is  8000 ;  the  second  is 
1200X3.     What  is  the  third  term? 

Observe  that  1200  is  3  times  the  square  of  the  first 
term;  consequently,  one  third  of  it  is  the  square  of 
the  first  term. 

25.  What  is  the  fourth  term  in  the  above  cube? 
What  is  the  root? 


SECTION    XX. 

RATIO.  —  PROPORTION. 

If  we  compare  the  two  numbers  3  and  9  in  order  to 
ascertain  their  relative  magnitude,  we  may  subtract  3 
from  9.     We  find  the  difference  to  be  6. 

There  is  another  way  of  comparing  the  two  num- 
bers. We  may  see  how  many  times  3  will  go  in  9. 
We  shall   find  the  quotient  to  be  3. 

The  numbers  we  obtain  in  each  of  these  compar- 
isons is  called  the  ratio  of  the  two  numbers ;  but  they 
differ  in  kind.  The  former  is  called  arithmetical  ratio ; 
the  latter,  geometrical  ratio. 

Arithmetical  ratio,  then,  expresses  the  difference  of 
two  numbers ;  geometrical  ratio  expresses  the  quotient 
of  one  of  the  numbers  divided  by  the  other.  As  we 
shall  speak  only  of  geometrical  ratio  in  what  follows 
here,  the  word  ratio,  whenever  it  is  used,  may  be  un- 
derstood to  mean  geometrical  ratio.  The  ratio  of  4 
to  2,  written  4:2,  is  2;  for  2  will  go  in  4  twice. 
>The  ratio  of  12  to  3,  written  12  :  3,  is  4;  for  3 
will  go  in  12,  4  times. 

I 


130 


MENTAL    ARITHMETIC. 


The  two  numbers  compared  are  together  called  the 
terms  of  the  ratio,  or  simply  the  ratio.  The  first  is 
called  the  antecedent ;  the  second  is  called  the  conse- 
quent. These  two  terms,  you  will  perceive,  corre- 
spond exactly  to  the  numerator  and  denominator  of  a 
fraction ;  for,  in  a  fraction,  the  numerator  is  divided 
by  the  denominator.  A  ratio  is,  then,  another  way  of 
expressing  a  fraction.  The  antecedent  is  the  numer- 
ator ;  the  consequent,  the  denominator.  4:2  is  the 
same  as  f ;    6:3  the  same  as  f . 

As  a  ratio  is  essentially  the  same  as  a  fraction, 
every  thing  is  true  of  a  ratio  which  is  true  of  a 
fraction. 

1.  What  effect  will  it  have  on  the  value  of  the 
ratio,  if  you  increase  the  antecedent  ?  If  you  diminish 
the  antecedent  ?  If  you  double  the  antecedent  ?  If 
you  divide  the  antecedent  by  2  ? 

2.  What  effect  will  it  have  on  the  value  of  the 
ratio,  if  you  increase  the  consequent  ?  If  you  diminish 
the  consequent  ?  If  you  multiply  the  consequent  ? 
If  you  divide  the  consequent  ? 

3.  Take  the  ratio  4  :  2.  How  can  you  multiply 
it  by  2  ?     In  what  other  way  ? 

4.  How  can  you  divide  it  by  2  ?  In  what  other- 
way  ? 

5.  Take  the  ratio  6  :  3.  How  can  you  multiply  it 
by  2  ?  Can  you  do  it  in  more  than  one  Avay  ?  If  you 
cannot,  why  ? 

6.  How  can  you  divide  it  by  4?  Can  you  do  it  in 
more  than  one  way  ?     If  not,  why  ? 

Take  the  ratio  4  :  2.  Multiply  both  terms  by  the 
same  number,  3,  for  example.  It  will  be  12  :  6.  You 
see  the  value  is  not  altered.  —  Divide  both  terms  4  :  2 
by  2.  It  will  be  2:1.  The  value  is  not  altered.  It 
is  still  2. 


PROPORTION.  131 


PROPORTION. 


If  there  are  four  numbers,  and  the  first  has  the  same 
ratio  to  the  second  that  the  third  has  to  the  fourth,  the 
four  numbers  are  said  to  be  in  proportion.  Thus  the 
numbers  2  :  1 : :  12  :  6  are  in  proportion.  The  first 
has  the  same  ratio  to  the  second,  that  the  third  has  to 
the  fourth.     The  ratio  is  2. 

A  proportion,  then,  is  the  equality  of  two  ratios. 

The  four  dots  : :  between  the  two  ratios,  are  the 
same  as  the  sign  of  equality,  =. 

In  order  to  preserve  the  proportion,  the  two  ratios 
must  always  be  equal  to  each  other.  You  may  make 
any  change  you  please  in  the  terms,  provided  you  do 
not  destroy  this  equality. 

Let  us  take  the  proportion  4  :  2  : :  12  :  6.  The 
value  of  the  two  ratios  is  now  equal. 

1st.  Multiply  the  antecedents  by  2.  8  :  2  : :  24  :  6. 
The  numbers  are  still  in  proportion,  for  the  value  of 
the  two  ratios  is  equal. 

2d.  Divide  the  antecedents  by  2.  2  :  2  : :  6  :  6. 
The  value  of  the  two  ratios  is  equal. 

3d.  Multiply  the  consequents  by  2.  4  :  4  : :  12  :  12. 
The  value  of  the  two  ratios  is  equal. 

4th.  Divide  the  consequents  by  2.  4  ;  1  : :  12  :  3. 
The  value  of  the  ratios  is  still  equal. 

5th.  Multiply  both  terms  of  the  first  ratio  by  2. 
8  :  4  : :  12  :  6  ;  or  multiply  the  2  terms  of  the  second 
ratio  by  2  ;  4  :  2  : :  24  :  12  ;  the  ratios  are  still  equal. 
In  the  same  way  we  might  take  any  other  number  for 
our  operations  instead  of  2.  The  same  operations 
might  be  performed  without  destroying  the  proportion. 

The  two  middle  terms  of  a  proportion  are  called  the 
means  ;  the  first  and  last  terms  are  called  the  extremes. 

In  a  proportion,  the  product  of  the  two  means  is 
equal  to  the  product  of  the  two  extremes. 


132 


MENTAL    ARITHMETIC. 


Take  the  proportion  4  :  2  : :  6  :  3.  The  product  of 
the  means,  2X6,  is  12;  and  the  product  of  the  ex- 
tremes, 4X3,  is  12. 

Take  the  proportion  6  :  2  :  :  9  :  3.  2x9=  18, 
3X6=18. 

Take  the  proportion  10  :  2  : :  30  :  6.    2X30  =  6X10. 

If  we  know,  then,  the  product  of  the  means,  we 
know  the  product  of  the  extremes. 

In  a  certain  proportion,  the  product  of  the  means  is 
30.     What  must  be  the  product  of  the  extremes  ? 

6.  Further,  if  we  know  the  product  of  the  means, 
and  if  we  know  one  of  the  extremes,  we  can  find  the 
other.  If,  as  in  the  above  case,  the  product  of  the 
means  is  30,  and  if  one  of  the  extremes  is  3,  what 
must  be  the  other  ? 

How  do  you  find  that  number  ? 

7.  If  the  product  of  the  means  is  30,  and  one  of  the 
extremes  is  10,  what  must  the  other  be  ? 

How  do  you  find  the  number  ? 

8.  If  the  product  of  the  means  is  30,  and  one  of  the 
extremes  is  5,  what  is  the  other  ? 

If  one  of  the  extremes  is  6,  what  is  the  other  ? 

If  one  of  the  extremes  is  15,  what  is  the  other? 

You  see,  therefore,  that,  if  you  multiply  the  means 
together,  and  divide  the  product  by  one  extreme,  the 
quotient  will  be  the  other  extreme. 

9.  If  the  product  of  the  means  is  72,  and  one  of  the 
extremes  is  24,  what  will  the  other  be  ? 

10.  If  the  first  three  terms  of  a  proportion  are 
9  :  6  : :  12,  what  must  the  fourth  term  be  ? 

11.  What  is  the  fourth  term  of  the  proportion 
5:3::  15? 

12.  Complete  the  proportion  8  :  6  : :  12. 

13.  Complete  the  proportion  14  :  8  : :  7. 

14.  Complete  the  proportion  10  :  4  : :  15. 

By  .ans  of  this  rule,  many  interesting  questions 
may       solved. 


PROPORTION.  133 

15.  If  8  yards  of  cloth  cost  6  dollars,  what  will  20 
yards  of  the  same  cloth  cost  ? 

It  is  evident  that  the  length  of  the  shorter  piece  is 
to  the  length  of  the  longer,  as  the  cost  of  the  shorter 
is  to  the  cost  of  the  longer.  Now,  we  know  all  these 
numbers  except  the  last,  and  can  express  them  in  the 

Yds.  Yds.      Dols. 

form  of  a  proportion,  thus,  8  :  20  : :  6.  8  is  the  length 
of  the  shorter  piece ;  20,  the  length  of  the  longer ;  6 
is  the  cost  of  the  shorter  piece.  The  fourth  term,  that 
is,  the  cost  of  the  longer  piece,  we  have  not  yet  found. 
You  must  discover  that  yourself.  How  can  you 
do  it? 

16.  If  18  yards  of  cloth  cost  15  dollars,  what  will 
12  yards  of  the  same  cloth  cost  ? 

What  do  you  here  seek,  —  the  quantity  of  cloth,  or 
the  price  ?  Is  it  the  price  of  the  longer,  or  of  the 
shorter  piece  ? 

How  can  you  make  a  proportion  with  the  two  quan- 
tities of  cloth,  and  the  two  sums  they  cost  ?  State 
this  proportion  in  general  terms,  putting  the  thing 
sought  as  the  fourth  term.  State  the  proportion  in 
figures,  all  except  the  fourth  term.  How  will  you 
find  the  fourth  term  ? 

17.  If  5  yards  of  cloth  cost  2  dollars,  what  will  7 
*  yards  of  the  same  cloth  cost  ? 

State  the  proportion  in  general  terms. 
State  the  first  three  terms  in  figures,  and  find  the 
fourth. 

18.  If  a  horse  travels  16  miles  in  3  hours,  how  far 
will  he  travel  in  2  hours  ? 

As  the  longer  time  is  to  the  shorter  time,  so  is  the 
greater  distance  to  the  smaller  distance. 

Remember  that  things  of  the  same  kind  should  stand 
t     in  the  same  ratio ;  and  that  the  quantity  sought  must 
be  the  fourth  term.     Then  inquire  what  the  true  pro- 
12 


134  MENTAL    ARITHMETIC. 

portion  must  be,  and  state  it  in  general  terms,  repeating 
the  trial,  if  necessary,  till  you  perceive  that  you  are 
right.  This  is  far  better  than  any  special  rule,  for  it 
leads  you  to  reason  on  what  you  do. 

19.  If  a  certain  number  of  cubic  feet  of  timber 
weighs  a  certain  number  of  hundred  weight,  and  if 
we  wish  to  know,  without  weighing,  how  many 
hundred  weight  a  certain  smaller  number  of  cubic 
feet  will  weigh,  what  will  be  the  proportion  in  general 
terms  ? 

20.  If  16  cubic  feet  of  wood  weigh  5  cwt.,  what 
will  6  cubic  feet  weigh  ? 

21.  If  3  barrels  of  flour  last  a  family  7  months,  how 
many  barrels  will  last  them  12  months  ? 

22.  If  an  iron  rod,  of  equal  size  throughout,  and  of 
a  certain  length,  weighs  a  certain  number  of  pounds, 
and  is  broken  into  two  parts,  not  in  the  middle,  how 
can  you  find  the  weight  of  one  of  the  parts  without 
weighing  it  ? 

23.  If  an  iron  roa*  7  feet  long  weighs  20  pounds, 
what  will  5  feet  of  it  weigh  ? 


COMPARISON   OF   SIMILAR   SURFACES. 

As  all  the  above  questions  may  be  answered  by 
analysis  as  well  as  by  proportion,  the  rule  of  propor- 
tion might  be  dispensed  with  for  the  purpose  of  solving 
this  kind  of  questions.  It  has,  however,  very  impor- 
tant and  interesting  applications  in  the  measurement  of 
similar  surfaces  and  solids.  To  prepare  for  this,  you 
must  attend  carefully  to  a  few  introductory  statements. 

Two  surfaces  are  similar  to  each  other  when  they 
are  shaped  alike,  though  they  may  be  unequal  in  size. 
Thus  a  large  circle  is  similar  to  a  small  circle,  for  they 
are  both  shaped  alike. 

So  one  square  is  similar  to  another,  though  they 


COMPARISON    OF     SIMILAR    SURFACES.  135 

may  be  unequal  in  size.  One  equilateral  triangle  is 
similar  to  another  equilateral  triangle. 

If  a  rectangle  is  twice  as  long  as  it  is  wide,  and  a 
larger  or  a  smaller  rectangle  is  twice  as  long  as  it  is 
wide,  the  two  are  similar.  In  the  same  way  any  two 
surfaces,  however  irregular  their  shape,  are  similar, 
provided  they  are  shaped  alike. 

We  will  now  come  to  a  stricter  definition  of  similar 
surfaces.  Similar  surfaces  are  such  as  have  their  cor- 
responding dimensions  proportional.  —  Take  the  circle. 
The  dimensions  of  the  circle  are  the  diameter  and  the 
circumference.  The  diameter  of  one  circle  is  to  its 
circumference,  as  the  diameter  of  a  larger  or  a  smaller 
circle  is  to  its  circumference.  For  you  have  learned 
before  that  the  circumference  of  a  circle  is  3|  times  its 
diameter. 

Take  the  square.  One  side  of  a  square  is  to  another 
side  of  it,  as  one  side  of  a  larger  or  smaller  square  is  to 
another  side  of  it. 

If  a  rectangle  is  twice  as  long  as  it  is  wide,  another 
rectangle,  in  order  to  be  similar,  whatever  be  its  size, 
must  be  twice  as  long  as  it  is  wide. 

1.  There  are  two  similar  rectangles.  One  is  8  feet 
long  and  6  feet  wide.  The  other  is  6  feet  long.  How 
wide  is  it  ? 

2.  There  are  two  similar  rectangles.  One  is  5  feet 
long  and  2  feet  wide.  The  other  is  11  feet  long.  How 
wide  is  it  ? 

3.  There  are  two  similar  right-angled  triangles. 
In  the  larger,  the  base  is  9  ;  the  perpendicular,  4 ;  —  in 
the  smaller,  the  base  is  8.  How  long  is  the  perpen- 
dicular ? 

We  will  now  come  to  the  comparison  of  the  areas 
of  similar  surfaces. 

4.  There  are  two  circles.  One  is  1  foot  in  diameter  ; 
the  other,  2  feet.  How  much  greater  is  the  area  of  the 
larger  than  the  area  of  the  smaller  ? 


136  MENTAL    ARITHMETIC. 

It  is  clearly  more  than  twice  as  large ;  for  you  could 
lay  two  of  the  smaller  circles  on  the  larger,  and  still 
leave  a  considerable  space  uncovered.  Before  answer- 
ing this  question,  we  will  take  the  simple  case  of  two 
squares,  one  of  which  measures  1  foot  on  a  side,  and 
the  other  2  feet.  You  perceive  the  larger  one  is  4 
times  as  great  as  the  smaller. 

Let  one  square  measure  2  feet  on  a  side;  the  other, 
4  feet.  How  much  greater  is  the  larger  than  the 
smaller?  The  smaller  contains  4  square  feet  ;  the 
larger,    16;    it  is  therefore  4  times  as  large. 

5.  If  one  square  measures  3  times  as  much  on  a 
side  as  another,  how  much  greater  is  its  area  than 
that   of  the   smaller? 

Let  one  square  measure  1  foot  on  a  side ;  the  other, 
3  feet.  In  what  ratio  are  their  areas? — Let  one  meas- 
ure 2  feet  on  a  side ;  the  other,  6.  In  what  ratio  are 
their  areas? 

The  area  of  the  larger,  you  find,  is  9  times  as  great 
as  that  of  the  smaller.  This  may  serve  to  suggest  the 
principle  by  which  the  areas  of  all  similar  surfaces 
may  be  compared. 

The  areas  of  similar  surfaces  are  to  each  other 
as  the  squares  of  their  corresponding  dimensions. 

Let  one  square  measure  1  foot  on  a  side;  another, 

2  feet.     I2  :  22  : :  1 :  4. 

Let  one  square  measure  2  feet,  and  another,  4  feet, 
on  a  side.     22  :  42  : :  4  :  16. 

Let  one  square  measure  1  foot  on  a  side  j   another, 

3  feet.      12:32::1  :  9. 

Let  one  square  measure  2  feet  on  a  side ;  another, 
6  feet.     22  :  62  : :  4  :  36. 

This  principle  applies  to  circles,  triangles,  and  all 
similar  surfaces  whatever.  You  can  now  recur  to 
question   4,  and  find  the  answer  to  it. 


COMPARISON    OF    SIMILAR    SOLIDS,  137 

6.  There  are  two  circles.  The  diameter  of  the 
greater  is  3  times  that  of  the  smaller.  The  area  of 
the  smaller  is  1  acre.  What  is  the  area  of  the 
greater  ? 

7.  There  are  two  circles.  The  diameter  of  the 
smaller  is  two  thirds  that  of  the  greater,  and  the  area 
of  the  smaller  is  4  acres.  What  is  the  area  of  the 
greater  ? 

8.  There  are  two  similar  triangles.  The  corre- 
sponding dimensions  are  as  3  to  4,  and  the  greater 
contains  12   acres.     What  does  the  smaller  contain? 

9.  A  farmer  fenced  a  triangular  piece  of  ground  for 
a  field ;  but,  finding  it  not  large  enough,  he  enlarged 
it,  making  each  side  one  third  greater  than  before,  and 
it  then  contained  5  acres.  How  much  did  it  contain 
at  first? 

10.  There  is  an  irregular  field  containing  8  acres. 
One  of  the  sides  measures  20  rods.  If  the  field  be 
enlarged,  retaining  the  same  form,  so  that  the  above- 
named  side  measures  25  rods,  how  much  land  will 
it  contain? 

11.  There  are  2  circles.  The  smaller  is  3  rods,  the 
larger  7  rods,  in  diametef.  How  much  greater  in  pro- 
portion is  the  area  of  the  latter  than  that  of  the  former  ? 

12.  There  are  2  circles,  one  with  a  diameter  of  3 
feet,  the  other  of  8.  How  much  greater  is  the  area 
of  the  larger  than  that  of  the  smaller? 


COMPARISON   OF   SIMILAR   SOLIDS. 

We  now  come  to  the  comparison  of  similar  solids. 

1.  Let  there  be  2  cubes,  one  of  them  measuring  1 
inch  on  a  side,  the  other  2  inches.     How  much  greater 

is  one  than  the  other  ? 
i 

You  will  perceive,  by  thinking  of  the  construction 
12* 


138  MENTAL    ARITHMETIC. 

of  the  cube,  that  the  cube  measuring  2  inches  has  in 
it  8  cubic  inches,  an<j  is  therefore  8  times  as  great  as 
the  one  measuring  only  1  inch. 

2.  Take  cubes  measuring  1  inch  and  3  inches. 
How  much  greater  is  the  latter  than  the  former? 

3.  Let  one  measure  1  inch,  tliQ  other  4  inches. 
How  much  greater  will  the  larger  cube  be? 

These  examples  may  suggest  the  principle  on  which 
all  similar  solids  are  compared. 

Similar  solids  are  to  each  other  as  the  cubes  of 
their  corresponding  dimensions. 

Take,  now,  the  first  of  the  above  three  examples. 
The  ratio  of  the  corresponding  dimensions  is  2:1; 
the  cubes  of  these  terms,  or  23  :  l3,  are  8:1;  and  this 
is  the  proportion  of  the  one  solid  to  the  other. 

In  the  second  example,  the  ratio  of  the  correspond- 
ing dimensions  is  3  : 1 ;  the  cubes  of  these,  33  :  l3,  are 
27  :  1 ;  and  this  is  the  ratio  of  the  two  solids. 

In  the  third  example,  the  ratio  of  the  corresponding 
dimensions  is  4 :  1 ;  the  cubes  of  these  terms,  43  :  l3, 
are  64 : 1,  which  is  the  ratio  of  the  two  solids  to  each 
other. 

4.  There  are  2  iron  balls.  The  smaller  is  1  inch, 
the  other  5  inches,  in  diameter.  How  much  does  the 
larger  weigh  more  than  the  smaller  ? 

5.  There  are  2  iron  balls.  Their  diameters  are  2 
inches  and  3  inches.  What  is  the  ratio  of  their 
weight  ? 

6.  If  the  diameter  of  2  balls  is  respectively  3  inches 
and  4  inches,  what  is  the  ratio  of  their  weight  ? 

7.  If  a  cubic  inch  of  stone  weigh  1  ounce,  how 
many  ounces  would  a  cubic  stone,  measuring  10 
inches,  weigh?  f 

8.  How  many  ounces,  if  the  cube  measured  11 
inches  ? 


COMPARISON    OF    SIMILAR    SOLIDS.  139 

9.  How  many  ounces,  if  the  cube  measured  12 
inches  ? 

10.  If  there  were  a  smaller  pyramid,  of  the  same 
material  and  shape  with  the  great  pyramid  of  Egypt, 
and  of  xV  its  height,  how  many  such  would  it  take  to 
equal  in  solid  contents  the  great  pyramid  ? 

11.  A  common  brick  weighs  4  pounds,  and  is  8 
inches  in  length.  How  much  will  a  similarly-shaped 
brick  weigh,  that  measures  16  inches  in  length? 

12.  If  an  axe  4  inches  wide  weighs  4|  pounds, 
what  will  be  the  weight  of  a  similar  axe  5  inches 
wide  ? 

13.  If  a  blacksmith's  anvil  1  foot  long  weighs  200 
pounds,  how  much  will  a  similar  anvil  weigh  that  is  2 
feet  long  ? 

14.  A  farmer  sells  2  stacks  of  hay  of  the  same 
shape  and  solidity.  The  smaller  is  10  feet  high,  and 
is  found  to  weigh  3  tons.  The  larger  is  15  feet  high. 
How  can  its  weight  be  determined,  without  weighing 
it  ?    and  what  will  the  weight  be  ? 

15.  There  are  2  similar  cisterns.  The  smaller  is  6 
feet  deep,  and  holds  500  gallons.  The  larger  is  8  feet 
deep.     How  many  gallons  will  it  contain  ? 

These  operations  will  be  rendered  more  easy,  if,  in 
every  case,  where  the  ratios  may  be  reduced,  you  re- 
duce them  to  their  lowest  terms. 

16.  If  a  coal-pit  8  feet  high  has  required  10  cords 
of  wood,  how  much  wood  would  be  required  for  a 
coal-pit  of  similar  shape  10  feet  high  ? 

17.  If  there  are  two  trees  shaped  alike,  the  smaller 
measuring  4  feet  in  circumference,  the  larger  5  feet, 
how  will  the  amount  of  wood  in  the  one  compare 
with  that  in  the  other  ? 

The  principle  given  above  applies  to  all  similar 
solids,  whether  bounded  by  plain  surfaces  or  by 
curved  surfaces. 


140  MENTAL    ARITHMETIC. 

18.  If  a  dwarf  measures  2  feet  in  height,  and  a 
man  of  the  same  form  and  solidity,  6  feet  high,  weighs 
180  pounds,  how  many  such  dwarfs  would  equal  the 
weight  of  the  man?     What  would  the  dwarf  weigh? 

19.  If  a  man  6  feet  2  inches  in  height  weighs  200 
pounds,  what  would  be  the  weight  of  a  giant  of  equal 
solidity  and  similar  form,  9  feet  3  inches  in  height  ? 

20.  If  an  animal  4  feet  high  weighs  600  pounds, 
what  will  an  animal  of  the  same  form  and  equal 
solidity  weigh,  whose  height  is  5  feet? 

21.  An  artist  in  Europe  has  made  a  perfect  model 
of  St.  Peter's  Church  at  Rome,  representing  every 
part  in  exact  proportion,  on  a  scale  of  1  foot  to  100 
feet.  If  the  material  of  the  model  is  of  the  same 
solidity  with  that  of  the  church,  how  many  times 
greater  is  the  solid  contents  of  the  church  than  that 
of  the  model? 

22.  If  a  granite  obelisk  were  constructed  in  the 
precise  form  of  the  Bunker  Hill  monument,  of  one 
tenth  its  height,  how  many  such  obelisks  would  the 
monument  furnish  material  to  construct? 

The  comparison  of  similar  surfaces  and  solids  by 
proportion  has  various  interesting  applications  in  de- 
termining the  comparative  strength  of  timbers  and 
materials  used  in  building  and  in  other  arts. 

Case  First.  —  The  strength  of  materials  to  resist  a 
strain  lengthwise. 

1.  If  an  iron  rod  half  an  inch  in  diameter  will  hold 
a  certain  weight  suspended  by  it,  how  much  greater 
weight  will  a  rod  hold  that  is  1  inch  in  diameter? 

Here  the  strength  is  in  proportion  to  the  size,  with- 
out regard  to  the  length  j  that  is,  as  the  square  of  the 

diameters. 

« 

2.  If  an  iron  rod  half  an  inch  in  diameter  will  sus- 


STRENGTH    OF    MATERIALS.  141 

pend  2  tons,  what  weight  will  a  rod  suspend  that  is 
three  fourths  of  an  inch  in  diameter? 

3.  A  builder  finds  that  an  iron  rod  1  inch  in  diam- 
eter will  suspend  a  certain  weight.     He  wishes,  how- 

;ever,  to  add  to  the  weight  half  as  much  more,  and, 
I  in  order  to  support  it,  substitutes  for  the  inch  rod 
.another  rod  1|  inches  in  diameter.  Will  it  sustain 
I  the  required  weight  ? 

4.  There  are  two  ropes  of  the  same  material ;  one, 
i  1£  inches  in  diameter  j  the  other,  2  inches.     What  is 

the  ratio  of  their  strength  ? 

Case  Second.  —  The  strength  of  beams  to  resist 
fracture  crosswise.  In  beams  of  the  same  material, 
length,  and  width,  but  of  different  depth,  the  strength 
varies  as  the  square  of  the  depth. 

1.  There  are  two  beams  of  equal  length ;  but  the 
depth  of  one  is  10  inches;  of  the  other,  12  inches. 
What  is  the  ratio  of  their  strength  ? 

2.  There  is  a  stick  of  timber  4  inches  thick  and 
12  inches  deep.  If  sawed  into  three  4-inch  joists, 
what  part  of  the  former  strength  of  the  whole  stick, 
when  placed  edgewise,  will  each  part  possess,  allow- 
ing nothing  for  waste  in  sawing? 

3.  There  is  a  stick  of  timber  10  inches  in  depth. 
If  4  inches  of  its  depth  be  removed,  what  will  be  its 
strength  compared  to  what  it  was  before  ? 

4.  There  are  two  sticks  of  timber  equal  in  length 
and  width ;  one,  7  inches  deep ;  the  other,  5.  What 
is  the  ratio  of  their  strength  ? 

5.  If  a  stick  of  timber  6  inches  deep  have  2  inches 
of  the  depth  removed,  will  it  be  weakened  more  than 
one  half? 

What  is  the  exact  ratio  of  its  present,  compared 
w%ith  its  former  strength  ? 

6.  A  builder  went   to  a  lumber-yard,  wishing   to 


142  MENTAL    ARITHMETIC. 

obtain  an  oak  beam  5  inches  wide  and  10  inches 
deep.  The  lumber-merchant  said,  "  I  have  not  such 
a  stick  ;  but  I  have  two  oak  sticks  of  the  right  length 
and  width,  and  7  inches  deep.  They  will  both, 
placed  side  by  side,  be  stronger  than  one  beam  10 
inches  deep."     "  Not  so  strong,"  said  the  builder. 

Which    was    right?      And   what    is    the   ratio   of 
strength  in  the  two  cases? 


NOTES    TO    PART    FIRST 


Note  1.  —  Page  16. 


This  exercise  should  be  often  reviewed,  till  the  pupils  can 
go  through  it  with  ease,  and  without  mistake.  No  exercise 
can  be  devised  that  will  more  rapidly  increase  the  learner's 
powers  in  addition. 

Note  2.  —  Page  17. —  To  the  Instructor. 

The  word  complement  means,  something  to  fill  up.  In 
arithmetic,  the  complement  of  a  number,  strictly  speaking, 
is  that  number  which  must  be  added  to  it,  to  make  it  up  to 
the  next  higher  order.  The  complement  of  a  number  con- 
sisting of  units  only,  as  3,  7,  9,  is  the  number  that  must  be 
added  to  make  it  up  to  10,  and  consists  of  units  only.  If  the 
number  consist  of  tens,  as  20,  50,  its  complement  is  the 
number  that  must  be  added  to  make  a  hundred,  and  consist 
of  tens.  If  the  number  is  hundreds,  its  complement  is  so 
many  hundreds  as  will  make  up  a  thousand. 

If  the  number  consist  of  several  orders,  its  full  comple- 
ment will  consist  of  the  same  orders,  of  such  an  amount  as  to 
raise  the  sum  to  the  next  order  above  the  highest  named  in  it. 
The  complement  of  745  is  255,  for  745  -\- 255 = 1000,  which 
is  the  order  next  above  the  highest  named  in  the  given  sum. 

The  more  restricted  use  of  the  word,  as  employed  in  the 
text,  is  sufficient  for  the  purposes  here  had  in  view. 

A  few  suggestions  will  here  be  made  in  reference  to  the 
best  mode  of  conducting  the  accompanying  recitation.  The 
object  of  the  lesson  is  to  cultivate  the  power  of  instantly 
associating  a  number  and  its  complement  together.  In  con- 
ducting the  recitation,  the  answer  to  each  question,  as  it  is 
given  out,  should   be  required  simultaneously  of  the  whole 


144 


NOTES    TO    PART    FIRST. 


class.  The  teacher  should  stand  before  them,  and  require 
that  every  eye  be  fixed  on  him.  The  questions  should  not 
be  hurried,  but  the  class  should  be  encouraged  to  answer 
instantly  on  hearing  the  question.  This  will  be  easy  in  the 
first  class  of  numbers  given,  which  are  even  tens.  In  regard 
to  the  remaining  numbers,  however,  which  are  not  even  tens, 
something  more  will  be  necessary.  Suppose  the  question  is, 
What  is  the  complement  of  37?  It  may  be  conducted  as 
follows : 

Teacher.     What  is  the  complement  of 30  ? 

Class.     70. 

Teacher.     Now,  listen  to  me  without  speaking.     What  is 

the  complement  of  30 ?  .   You  observe,  I  am  going  to 

say  something  more.     What  will  it  be  ? 

Class.     Something  between  30  and  40. 

Teacher.  Well,  then,  whereabouts  will  the  complement 
be  found  ? 

Class.     Between  60  and  70. 

Teacher.  Very  good.  Now,  when  I  say  30,  and  keep 
my  voice  suspended,  showing  that  that  is  not  all,  what 
number  can  you  think  of,  that  you  know  will  be  a  part 
of  the  complement? 

Class.     60. 

Teacher.  Very  well.  Now,  listen.  What  is  the  comple- 
ment of  30 ?     What  have  you  now  in  your  mind? 

Class.     60. 

Teacher.  Well.  Now,  once  more  listen,  and  all  answer 
as  soon  as  you  hear  the  question.  What  is  the  comple- 
ment of  37  ? 

Class.     63. 

In  the  following  questions,  let  the  teacher  always  make  a 
short  pause  between  pronouncing  the  tens  and  the  units ; 
and  if  the  class  hesitate  or  disagree  in  their  answer,  let  the 
question  be  resolved  into  its  elements,  and  each  one  pre- 
sented separately.  Thus,  if  64  is  the  number,  and  the  class 
have  not  answered  promptly  and  alike,  say  thus;  "  What  is 
the  complement  of  60?" 

Class.     40. 

Teacher.     What  is  the  complement  of  60 ?     What 

do  you  think  of?  t 

Class.     30. 


NOTES    TO    PART    FIRST.  145 

Teacher.  Now,  answer  all  together.  What  is  the  com- 
plement of  64? 

Class.     36. 

In  the  examples  of  addition  that  follow,  the  teacher  should 
make  a  pause  between  the  two  numbers,  and  see  that  every 
member  of  the  class  is  intent  and  eager  to  catch  the  second 
number,  and  answer  instantly.  A  few  questions  answered 
by  the  whole  class  in  this  way,  will  benefit  them  more  than 
whole  pages  recited  in  an  indolent  and  listless  manner. 


Note  3.  —  Page  17. 

In  these  and  all  other  examples,  the  large  numbers  should 
be  taken  first.  If  the  pupil  begins  with  the  units,  as  in 
written  arithmetic,  he  should  be  checked  at  once.  Such  a 
method  would  only  lead  to  a  laborious  imitation  of  the  pro- 
cess of  written  arithmetic,  which  is  not  the  natural  one,  and 
could  give  no  new  power  to  the  pupil,  nor  awaken  any  new 
interest  in  the  study.  Only  a  small  portion  of  these  ques- 
tions should  be  recited  at  one  lesson. 


Note  4.  —  Page  25. 

Care  must  be  taken  here,  that  the  pupil  does  not  imitate 
the  process  of  written  arithmetic,  but  be  required  to  re- 
gard every  number  in  its  true  value.  Thus,  in  the  ques- 
tion, "What  is  one  fifth  of  250  1 "  he  must  not  say,  "5 
in  25  is  contained  5  times ;  and  5  in  0,  no  times ;  "  but, 
"One  fifth  of  25  is  5;  therefore,  one  fifth  of  250  is  50." 


Note  5.  — Page  28. 

In  the  higher  as  well  as  the  lower  numbers,  let  the  pupil 
grapple  at  once  with  the  number  as  it  stands.  In  this  way 
his  interest  will  be  very  much  increased.  He  will  see, 
throughout,  the  progress  he  is  making.  Whereas,  in  written 
arithmetic,  as  usually  studied,  the  pupil  has  no  sooner  begun 
13  K 


146  NOTES    TO    PART    FIKST.  ' 

an  operation  than  he  loses  sight  of  the  process,  and  goes  on 
in  blind  bondage  to  his  rule,  till  he  comes  out  at  the  end, 
and  then  looks  to  the  book,  as  to  an  oracle,  for  the  answer. 

Let  the  oldest  class  in  arithmetic  in  a  school  be  called  up, 
and  one  of  them  be  required  to  perform  on  the  board  the 
question,  "What  is  one  sixth  of  43,248?"  and  when  he 
has  obtained  the  first  quotient  figure,  stop  him,  and  ask  him 
what  he  has  now  done.  He  will  most  likely  be  unable  to 
tell.  The  answer  he  will  give  will  probably  be,  that  he  has 
divided  43  by  6;  and  no  one  of  his  class  will  probably  have 
a  better  answer  to  offer.  If  he  says  he  has  divided  43 
thousand,  he  is  still  wrong;  for  he  has  divided  only  42 
thousand,  leaving  one  thousand  undivided. 

In  some  of  the  examples  given  in  this  section,  the  large 
numbers  may  be  separated  in  different  ways  preparatory  to 
division.  Thus,  in  the  last  example,  92,648  may  be  divided 
80,000,  12,000,  600,  48;  or  88,000,  4000,  640,  8;  and  in 
still  other  ways. 

Pupils  should  be  encouraged  to  exhibit  more  methods  than 
one  for  obtaining  the  answer.  If  a  scholar  has  two  methods, 
he  should  be  allowed  to  give  them  both ;  and  if  another  has 
a  different  one  still,  it  should  be  brought  forward ;  and  the 
most  lucid  and  easy  one  should  receive  the  commendation  of 
the  teacher. 

Note  6.  — Page  101. 

Strictly  speaking,  there  is  no  relation  in  quantity  between 
a  line  and  a  surface,  but  only  between  a  line  and  the  dimen- 
sions of  a  surface.  By  the  square  of  a  line  is  meant  a  square 
surface,  each  of  whose  sides  has  the  same  length  as  the 
given  line. 


PART    SECOND; 


CONTAINING 


RULES    AND    EXAMPLES    FOR    PRACTICE 


WRITTEN  ARITHMETIC. 


NUMERATION    OF   WHOLE    NUMBERS. 

In  common  arithmetic,  there  are  9  figures  used  for 
the  expression  of  numbers.  1,  one  ;  2,  two  ;  3,  three  ; 
4,  four ;  5,  five ;  6,  six  ;  7,  seven ;  8,  eight ;  9,  nine. 
When  one  of  these  figures  stands  alone,  it  signifies  so 
many  units,  or  ones ;  when  two  figures  stand  side  by- 
side,  the  left-hand  figure  signifies  so  many  tens ;  when 
three  stand  side  by  side,  the  left-hand  figure  signifies 
so  many  hundreds ;  and  universally,  as  you  advance 
to  the  left,  the  figures  increase  in  value  tenfold  at  each 
step,  as  will  be  seen  in  the  table  on  the  next  n*«r- 

The  right-hand  place  is  always  tha'  '  all*s\  w^n 
.,  °  j1  J.  ,^i,  0,  must  stand  u, 

there  are  tens,  and  no  units,  a  cipv    '    ;        ,TT^  tr.  no 
*i  •*»       i         .i_        r»rv      miiS  merely  serves  to  oc- 

the  unit's  place,    hus,  20      ^  *    figure,  2,  is 

cupy  the  una's  place,  an£  hun|re(te  and 

m  the  place  of  tens      "  d         T^  the 

no  tens  nor  units  cWU  ^1Pucl°  "^  '  Qnn  . 

unit's  place.  -^  one  in  the  place  of  tens;  as,  200, 
and  so  n*"  all  higher  numbers. 

To  annex  a  cipher  to  a  figure,  therefore,  is  the  same 
as  to  multiply  the  number  by  ten ;  for  it  removes  the 
figure  from  the  unit's  place  to  the  place  of  tens.  lo 
aiinex  two  ciphers,  is  the  same  as  to  multiply  the  num- 
ber by  a  hundred ;  for  it  removes  the  figure  from  the 
unit's  place  to  that  of  hundreds. 


j 


148 


TABLE    OF    NUMERATION. 


s'i 


two. 

two  tens,  that  is,  twenty. 

two  hundred. 

enumerate. 

enumerate. 

i  two  tens  of  thousands, 
\  that  is,  twenty  thousand. 

two  hundred  thousand. 

enumerate. 

twenty-two  million. 

enumerate. 

enumerate.  • 


In  writing  numbers,  every  place  not  occupied  by  a  figure* 
must  be  occupied  by  a  cipher.     Otherwise  the  true  value  of 


NUMERATION. 


149 


the  figures  at  the  left  hand  of  that  place  would  not  be  pre- 
served. Thus,  if  you  wish  to  write  in  figures  the  number 
three  hundred  and  four,  as  there  are  no  tens,  a  cipher  must 
stand  in  the  place  of  tens,  304.  Should  you  omit  the  cipher, 
and  write  14,  the  3  would  have  slid  into  the  tens'  place,  and 
it  would  not  express  three  hundred  and  four. 

As,  in  advancing  to  the  left,  figures  increase  their  value 
tenfold  at  each  step,  so,  if  you  begin  at  any  place  in  a  line  of 
figures,  and  move  towards  the  right,  the  figures  will  diminish 
in  value  tenfold  at  each  step :  that  is,  each  figure  will  signi- 
fy but  a  tenth  part  of  what  it  would,  if  it  stood  in  the  next 
left-hand  place.     This  will  prepare  you  to  look  at  the 

NUMERATION   OF   DECIMALS. 

Whole  Numbers.  Decimals. 


S   =     =    3- 


7th  place,  millions. 

9th  place,  tens  of  millions. 

9ih  plaoe,  lunula,  of  mills. 

r    5-    S- 
a"   m    5" 

i  3  s 
[ft 

S 
s 

tr 

3 

8 

IS 

J 

2 

1 

2 

4 

2 

f . 

1 

2 
2" 

2; 

3;- 

-i  £ 

i  § 

r  b 

'     1 

1    1 

ir  =~ 

I 

2 

7th  place,  tens  of  mill'ths. 
6th  place,  millionths. 
5th  place,  hund.  of  thou'ths. 
4th  place,  tens  of  thou'ths. 

1 

3 

6 

7  ] 

2; 
2; 
2^ 

L4 

16 

4 

>2 
>2 

[7 
J  9 

1 

17 

2 
2 
22 

7 
6742 

two,  and  two  tenths. 

5  four,   and  two   tenths,  and  five 

(    hundredths,  or  25  hundredths. 

{  twenty-two,   and  twenty- 
)        two  hundredths. 


two  thousandths, 
enumerate. 

four  hundred   and    seventeen 
hundred-thousandths. 


enumerate. 


13* 


150  ADDITION. 

SECTION    I. 

ADDITION. 

Addition  is  the  uniting  of  several  sums  into  one,  to 
show  their  amount. 

Rule.  —  Set  down  the  numbers,  units  under  units, 
tens  under  tens,  and  so  on.  Add  the  column  of  units; 
set  down  the  units  of  the  amount,  and  carry  the  tens,  if 
there  are  any,  to  the  column  of  tens.  Add  the  column 
of  tens,  and  set  down  the  unit  figure  of  the  amount,  car- 
rying the  figure  of  tens  to  the  next  column ;  and  so  on. 
In  adding  the  last  column,  set  down  the  whole  amount. 

To  prove  the  work,  repeat  the  operation,  beginning 
at  the  top,  and  adding  downwards. 


Examples. 


8.  871  +  934  +  340. 

9.  516  +  617  +  713. 

10.  685  +  937  +  742. 

11.  840  +  931  +  672. 

12.  963  +  847  +  784. 

13.  421  +  317  +  844. 


1.  472  +  842. 

2.  376  +  421  +  645. 

3.  431  +  843  +  794. 

4.  821  +  954  +  359. 

5.  267  +  549+121. 

6.  834  +  682  +  762. 

7.  468  +  912  +  683. 

14.  6342  +  1896  +  4741  +  8962 

15.  3249  +  856  +  8007  +  4990. 

16.  3819  +  42  +  906  +  1728. 

17.  1645  +  2718  +  92  +  1807. 

18.  1543  +  1899  +  3054  +  26. 

19.  1854+1962  +  2168  +  666. 

20.  1062  +  6300  +  9071  +  7001, 

21.  2593  +  1801  +  9201  +  2113. 

22.  9064  +  2118  +  1802  +  3076. 

23.  1001  +  9016  +  7990  +  26. 

24.  106  +  2307  +  9436+108. 

25.  1214  +  6403  +  7113  +  4009, 


ADDITION.  151 

26.  In  1840,  the  population  of  the  Ne\^  England 
States  was  as  follows:  Maine,  501,793;  New  Hamp- 
shire, 284,574  ;  Vermont,  291,948  ;  Massachusetts, 
737,699;  Connecticut,  309.978;  Rhode  Island,  108,850. 
What  was  the  population  of  all  the  New  England 
States  ? 

27.  The  population  of  the  Middle  States,  in  1840, 
was  as  follows :  New  York,  2,428,921 ;  New  Jersey, 
373,306;  Pennsylvania,  1,724,033;  Delaware,  78,085; 
Maryland,  469,232;  Virginia,  1,239,797.  What  was 
the  total  population  of  the  Middle  States  ? 

28.  The  population  of  the  Southern  States,  in  1840, 
was  —  North  Carolina,  753,419  ;  South  Carolina, 
594,398;  Georgia,  691,392  ;  Alabama,  590,756 ;  Ten- 
nessee, 829,210  ;  Mississippi,  375,651  ;  Arkansas, 
97,574;  Louisiana,  352,411.  What  was  the  total  pop- 
ulation of  these  states  ? 

29.  In  1840,  the  population  of  the  Western  States 
was  as  follows:  Ohio,  1,519,467;  Indiana,  685,866  ; 
Illinois,  476,183;  Michigan,  212,267;  Kentucky, 
777,828;  Missouri,  383,702.  What  was  the  total 
population  of  these  states  ? 

30.  What  is  the  total  population  of  all  the  United 
States,  as  set  down  in  the  four  preceding  examples  ? 

Whenever,  in  adding  a  column,  two  figures  occur 
together,  which  amount  to  10,  as  8  and  2,  7  and  3, 
take  them  both  together,  and  call  them  10.  This  will 
make  the  addition  more  rapid  and  easy. 

When  you  have  become  familiar  with  the  operations 
in  addition,  you  may  occasionally  vary  your  method, 
by  taking  two  columns  of  figures  at  a  time.  If  you 
have  been  thorough  in  the  mental  part  of  this  work, 
you  will  be  able  to  do  this.  It  will  furnish  an  agree- 
able variation  in  your  method  of  work,  and  greatly 
increase  your  power  of  rapid  calculation. 

31.  This  method  is  seen  in  the  following  example: 


152  SUBTRACTION. 

3124  ^      42  and  81  are  123,  and  24  are  147.     Set 
7681    I  down  the  47,  and  carry  the  1  hundred  to  the 
4942   .  column  of  hundreds.      50  and  76  are    126, 
15747  J   an(i  31  are  157. 

It  will  be  well  often  to  adopt  this  as  your  method 
of  proof.  After  performing  the  w^>rk  by  taking  one 
column  at  a  time,  prove  it  by  taking  two  columns ;  or 
perform  it  first  in  the  latter  way,  and  prove  it  in  the 
other. 

32.  1467  +  894  +  1721  +  8396. 

33.  9461  +  8134  +  2016  +  4317. 

34.  84161  +  9632  +  78167+43180. 

35.  109761+20671  +  437674  +  963. 

36.  26431  +  184097  +  467124  +  84321. 

37.  43126  +  91434  +  237210  +  127. 

38.  1235467+1096  +  34271  +  4081. 

39.  10467  +  31762  +  10921  +  9634. 

40.  37193+10634  +  206721  +  104367. 


SECTION   II. 

SUBTRACTION. 

Subtraction  is  the  taking  of  a  smaller  number  from 
*  a  larger,  to  show  the  difference.     The  larger  number 
is  called  the  minuend;  the  smaller,  the  subtrahend; 
the  difference  is  called  the  remainder. 

Rule.  —  Set  down  the  numbers,  the  larger  number 
uppermost,  units  under  units,  tens  under  tens.  Sub- 
tract the  units  of  the  lower  number  from  the  unit 
figure  above,  and  set  down  the  difference.  Proceed  in 
the  same  way,  with  the  tens  and  higher  orders,  to  the' 


SUBTRACTION. 


153 


close.  If,  in  any  case,  the  figure  of  the  minuend  is 
less  than  the  figure  below  it,  increase  it  by  10,  by  bor- 
rowing 1  from  the  next  higher  figure  of  the  minuend  ; 
remembering,  at  the  next  step,  that  the  figure  in  the 
minuend  has  already  been  diminished  by  1. 

To  prove  the  work,  add  the  remainder  and  the  sub- 
trahend together,  and,  if  the  work  is  correct,  the  sum 
will  agree  with  the  minuend. 

Examples. 


1. 

748  —  365. 

14. 

8990—7096. 

2. 

674—582. 

15. 

8243  —  6492. 

3. 

849—634. 

16. 

784—96. 

4. 

347—267. 

17. 

210  —  100. 

5. 

431  —  249. 

18. 

681  —  504. 

6. 

867—312. 

19. 

901  —  75. 

7. 

419—224. 

20. 

16432—14968. 

8. 

519  —  499. 

21. 

195864—137461. 

9. 

318—201. 

22. 

228476  —  13962. 

10. 

856  —  106. 

23. 

740016  —  116799 

11. 

3416—2999. 

24. 

86400  —  199. 

12. 

4162—4091. 

25. 

10006  —  4364. 

13. 

7089—3007. 

26.  America  was  discovered  in  1492.  Plymouth 
was  settled  in  1620.  How  long  was  that  after  the 
discovery  of  America? 

27.  The  Independence  of  the  United  States  was 
declared  in  1776.  How  long  was  that  after  the  settle- 
ment of  Plymouth  ? 

f  28.  George  Washington  was  born  in  1732.  He 
took  command  of  the  American  armies  in  1776.  How 
old  was  he  then  ? 

29.  General  Washington  became  President  of  the 
United  States  in  1789.     How  old  was  he  then  ? 

30.  In  1820,  the  population  of  Maine  was  298,335. 
^In  1830,  it  was  399,955.     What  was  the  increase  in 

10  years  ? 


L 

154  SUBTRACTION. 

31.  The  population  of  Maine,  in  1840,  was  501,973. 
What  was  the  increase  from  1830  to  1840  ? 

32.  The  population  of  Massachusetts,  in  1810,  was 
472,040 ;  in  1820,  523,487.  How  much  had  it  in- 
creased from  1810  to  1820? 

33.  The  population  of  Massachusetts,  in  1830,  was. 
610,408.     How  much  had  it  increased  from  1820  to 
1830? 

34.  In  1840,  the  population  of  Massachusetts  was 
737,699.  How  much  had  it  increased  from  1830  to 
1840? 

35.  The  population  of  the  state  of  New  York,  in 
1810,  was  959,949.  In  1820,  it  was  1,372,812.  What 
was  the  gain  ? 

36.  The  population  of  New  York,  in  1830,  was 
1,918,608.     What  was  the  gain  from  1820  to  1830  ? 

37.  In  -1840,  it  was  2,428,921.  What  was  the  gain 
from  1830  to  1840  ? 

38.  The  population  of  Ohio,  in  1810,  was  230,760. 
In  1820,  it  was  581,434.  What  was  the  gain  from 
1810  to  1820  ? 

39.  In  1830,  the  population  of  Ohio  was  937,903. 
What  was  the  increase  from  1820  to  1830? 

40.  In  1840,  the  population  of  Ohio  was  1,519,467. 
What  was  the  increase  from  1830  to  1840  ? 

Another  method  of  performing  subtraction,  often 
more  convenient  than  the  former,  is  the  follow- 
ing:— 

Regard  the  subtrahend  as  a  round  number,  one 
greater  than  the  figure  of  its  highest  order ;  that  is,  if 
the  subtrahend  is  43,  call  it  50;  if  251,  call  it  300. 
Subtract  this  round  number  from  the  minuend,  and 
then  to  the  remainder  add  the  complement  required 
to  make  up  the  subtrahend  to  the  round  number ;  as 
follows :  — 


MULTIPLICATION.  155 

41.  674—381.  400  from  674  leaves  274.  Add  19, 
the  complement  of  381.     274+19  =  293,  Ans. 

Apply  this  method  to  example  11,  above. 

42.  3416  —  2999.  The  first  remainder,  you  see,  is 
416,  and  the  complement  is  1.     Ans.  417. 

This  example  shows  how  much  shorter  the  work 
often  becomes  by  adopting  this  method. 

One  of  the  above  methods  may  be  used  as  a  proof  of 
the  other. 


43.  384—219. 

44.  1260—984. 


45.  1679—291. 

46.  2496  —  954. 


SECTION    III. 

MULTIPLICATION. 

In  multiplication,  a  number  is  repeated  a  certain 
number  of  times,  and  the  result  thus  obtained  is  called 
the  product. 

Rule.  —  Set  down  the  smaller  factor  under  the 
larger,  units  under  units,  tens  under  tens.  Begin  with 
the  unit  figure  of  the  multiplier.  Multiply  by  it,  first, 
the  units  of  the  multiplicand,  setting  down  the  units 
of  the  product,  and  reserving  the  tens  to  be  added  to 
the  next  product.  Proceed  thus  through  all  the  figures 
of  the  multiplicand.  If  there  are  more  figures  than 
one  in  the  multiplier,  take,  next,  the  tens,  and  multiply 
the  figures  of  the  multiplicand  as  before,  setting  the 
figures  of  the  product  one  degree  farther  to  the  left 
than  before. 

Add  the  several  partial  products,  and  the  amount 
will  be  the  whole  product. 


156 


MULTIPLICATION. 


1.  342X64. 


2. 
3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 


Examples. 

Thus,      342  i 
64 

1368 
2052 


346  X  34. 
579  X  82. 
976  X  38. 
826  X  91. 
376  X  121. 
345  X  243. 
798X114. 
6181X35. 
6821  X  82. 
7413  X  96. 
7921x22. 
8964  x  85. 
9056  X  43. 
8007X41. 
4559  X  741. 
9642  X  864. 
8721x317. 
1841  X  134. 


Ans.    21888  - 


20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 


Proof. 

The    best    practical 
"     method  of  proof  is 
carefully  to  repeat 
the  operation. 

13763  X  26. 
97623X318. 
1172671X216. 
1874215X341. 
742634X912. 
189423  X  62. 
14376281  X  194. 
17284265X36. 
671234X427. 
1895453  X  28.       . 
3469528  X  672. 
906421384  X  923. 
713489605  X  84. 
843469537  X  906. 
236749024X516. 
443754262X916. 
1123496113X413. 


37.  The  average  length  of  the  state  of  Massachu- 
setts is  150  miles  ;  its  breadth,  50  miles.  How  many- 
square  miles  does  it  contain  ? 

38.  The  average  length  of  Pennsylvania  is  275 
miles  ;  its  breadth,  165  miles.  How  many  square 
miles  does  it  contain? 

39.  The  state  of  Ohio  averages  223  miles  in  length, 
180  in  breadth.  How  many  square  miles  does  it  con- 
tain? 


MULTI  PLICATION.  157 

40.  The  state  of  Illinois  averages  245  miles  in 
length,  147  in  breadth.  How  many  square  miles 
does  it  contain? 

41.  If  there  are  365  days  in  one  year,  how  many 
days  are  there  in  25  years? 

42.  If  the  wages  of  a  soldier  is  8  dollars  a  month, 
what  will  be  the  wages  of  7867  soldiers  for  12  months  ?  $ 

43.  There  are  320  rods  in  1  mile.  How  many  rods 
are  there  in  278  miles  ? 


44.  741X84. 

45.  19643  XS92. 

46.  246731X9210. 


47.  946734X496. 

48.  1623X198. 

49.  9336X1998. 


When  the  multiplier  is  a  composite  number,  you 
may  multiply  first  by  one  of  its  factors,  and  the 
product  thus  obtained  by  the  other  factor,  or  by  the 
others  in  succession,  if  there  are  more  than  two. 

Apply  this  method  to  the  following  examples :  — 

50.    8476X45.       51.    1371X125.       52.    7465X108. 

If  a  figure  in  the  multiplier  is  a  factor  of  the  figure 
in  the  next  higher  place,  you  may  shorten  the  opera- 
tion by  multiplying  the  partial  product  of  the  lower 
figure  by  the  other  factor  of  the  higher.  '  Thus,  in 
example  44,  above,  having  found  4  times  741,  you 
know  that  8  times  the  same  is  twice  as  many,  and  80 
times  is  20  times  as  many.  You  need,  therefore,  only 
double  the  line  of  the  first  partial  product,  setting  it 
one  degree  farther  to  the  left,  to  express  the  tenfold 
higher  value.  The  same  may  be  done  if  the  right- 
hand  figure  is  a  factor  of  the  number  expressed  by 
the  next  two  higher  figures. 

Apply  the  process  to  the  following  examples:  — 


53. 

947X639. 

56. 

27934X369. 

54. 

13674X4812. 

57. 

67514X64164. 

55. 

19742X568. 

58. 

259385X13212 

If  the  multiplier  is  10,  or  any  power  of  10,  annex 
14 


158  DIVISION. 

to  the  multiplicand,  for  the  answer,  as  many  ciphers 
as  there  are  in  the  multiplier. 

If  the  multiplier  consists  of  9's,  add  as  many  ciphers 
to  the  multiplicand  as  there  are  9's  in  the  multiplier, 
and  from  the  product  subtract  the  multiplicand.  The 
remainder  will  be  the  product  sought ;  for,  by  adding 
the  ciphers,  you  multiply  by  a  number  greater  by  one 
than  the  multiplier.  The  multiplicand,  therefore,  will 
be  found  in  the  product  once  too  many  times.  So,  if 
the  multiplier  is  2  or  3  less  than  some  power  of  10, 
you  may  do  the  same,  remembering  to  take  the  multi- 
plicand out  as  many  times  as  the  multiplier  is  units 
less  than  a  power  of  10. 

In  this  way  perform  the  following  examples:  — 


59.  3847X99. 

60.  4572X999. 


61.  54327X98. 

62.  45314X997. 


SECTION     IV. 

DIVISION. 

In  division,  two  numbers  are  given,  in  order  to  find 
how  many  times  one  contains  the  other,  or  in  order 
to  separate  one  number  into  as  many  equal  parts  as 
there  are  units  in  the  other. 

The  number  to  be  divided  is  the  dividend;  the 
number  it  is  divided  by  is  the  divisor ;  the  answer 
is  the  quotient. 

To  perform  the  operation,  set  down  the  divisor  at 
the  left  of  the  dividend.  Take  as  many  figures  on 
the  left  of  the  dividend  as  will  contain  the  divisor 
one  or  more  times.  See  how  many  times  the  divisor 
is  contained  in  these  figures,  and  set  down  the  num- 
ber as  the  first  figure  of  the  quotient.     Multiply  the' 


DIVISION. 


159 


divisor  by  the  quotient  figure,  and  subtract  the  product 
from  the  number  taken.  To  the  remainder  bring 
down  another  figure  of  the  dividend ;  and  proceed  as 
before. 

1.    13276-^122;  thus, 

'  122 )  13276  (  108,  quotient. 
122 


1076 
976 


100,  remainder. 

To  prove  the  work,  multiply  the  divisor  and  the 
quotient  together,  and  add  the  remainder,  if  there  be 
any ;  and  the  amount,  if  the  work  be  right,  will  be 
equal  to  the  dividend. 

Thus,  in  the  above  example,     122 

108 

976 
122 

100 


13276 


11.  3059-^214. 

12.  700601  -f-  34. 

13.  643817-^150. 

14.  300796-^145. 

15.  3264291-^27. 

16.  18947633-^-181. 

17.  384628910 -r  26. 

18.  137900-^62. 

19.  3946908  -M 72. 

20.  A  man  divided  35,785  dollars  equally  among 
five  children.     How  much  did  each  receive? 

21.  In  one  barrel  of  flour  there  are  196  lbs.     How 
many  barrels  of  flour  are  there  in  13,916  lbs.  ? 

*  22.    In  1840,  the  population  of  Maine  was  501,793. 


2. 

11764-^34. 

3. 

47478-1-82. 

4. 

37088-^38. 

5. 

75116-^91. 

6. 

45496^-121. 

7. 

83835-^243. 

8. 

90972  —  114. 

9. 

89743  —  17. 

10. 

7426831  -t-141. 

160  DIVISION. 

The  state  contained  then  30,000  square  miles.  How 
many  inhabitants  were  there,  on  an  average,  to  a 
square  mile? 

23.  The  state  of  Massachusetts  contained,  in  1840, 
737,699  inhabitants.  Its  territory  is  7500  square  miles. 
How  many  inhabitants  are  there  to  a  square  mile  ? 

24.  The  population  of  Ohio,  in  1840,  was  1,519,467. 
Its  territory  is  40,000  square  miles.  How  many  in- 
habitants  to  a  square  mile? 

If  the  divisor  is  less  than  12,  the  multiplication  and 
subtraction  may  be  carried  on  in  the  mind,  and  only 
the  quotient  set  down.  This  may  most  conveniently 
be  written  directly  under  the  dividend. 

25.  7846  -=-  3.      Operation,  3  )  7846 

2615+1;  remainder. 
26.    964385 -=-5. 


27.  346218 -=-7. 

28.  214681 -=-9. 

29.  684219 -=-8. 

30.  9640279 -=-4. 


31.  146710063 -=-6. 

32.  1143762 -Ml. 

33.  1964217  -=-12. 

34.  4691382 -=-4. 


It  is  well  to  adopt  the  method  of  short  division 
sometimes  when  the  divisor  is  larger  than  12. 


35.  33467-=- 15. 

36.  46943 -M5. 

37.  81743  4-16. 


38.  91674 -=-21. 

39.  673845 -=-22. 


Miscellaneous  Examples  on  the  foregoing  Rules. 

1.  A  merchant  began  to  trade  with  4325  dollars. 
He  gained  in  one  year  784  dollars.  What  was  he 
then  worth? 

2.  A  man's  income  is  948  dollars  a  year.  His  ex- 
penses are  762  dollars.  How  much  does  he  save  of 
his  income  in  one  year? 

3.  How  much  will  he  save  in  9  years  ? 


MISCELLANEOUS    EXAMPLES.  161 

4.  A  man  bequeathed  his  property,  3882  dollars, 
one  third  to  his  wife,  and  the  remainder,  in  equal 
shares,  to  his  four  children.  What  was  each  child's 
share  ? 

5.  A  merchant  buys  643  barrels  of  flour  at  5  dol- 
lars a  barrel.  He  pays  in  addition,  for  freight,  65  dol- 
lars ;  for  insurance,  17  dollars.  What  does  the  whole 
cost  him  then  ?    What  does  each  barrel  cost  him  ? 

6.  A  drover  bought  7  oxen  for  46  dollars  a  head, 
12  cows  for  32  dollars  a  head,  96  sheep  for  3  dollars  a 
head.     How  much  did  they  all  come  to  ? 

7.  A  drover  buys  48  head  of  cattle  at  32  dollars  a 
head.  The  whole  expense  of  driving  them  to  market 
and  selling  them  is  72  dollars.  He  sells  them  for  38 
dollars  a  head.     What  does  he  gain  ? 

8.  A  laborer  receives  16  dollars  for  every  four 
weeks'  labor.  He  works  48  weeks.  What  do  his 
earnings  amount  to? 

9.  A  man  buys  7  tons  of  hay  in  the  field  for  13 
dollars  a  ton.  The  cost  of  carrying  it  all  to  market  is 
48  dollars.  He  sells  it  for  15  dollars  a  ton.  Does  he 
gain,  or  lose?    and  how  much? 

10.  A  man  receives  a  salary  of  950  dollars.  He 
spends  for  groceries  154  dollars;  for  milk,  21  dollars; 
for  meat,  75  dollars  ;  for  wood,  67  dollars ;  for  cloth- 
ing, 184  dollars ;  for  horse-hire,  38  dollars ;  for  jour- 
neying, 93  dollars ;  for  repairs,  19  dollars ;  for  hired 
help,  132  dollars ;  for  attendance  of  the  physician,  26 
dollars;  for  furniture,  51  dollars;  for  house  rent,  184 
dollars :  and  86  dollars  in  charity  and  other  incidental 
expenses.  Has  he  spent  more  than  his  salary,  or  less  ? 
and  how  much  ? 

11.  A  man  works  five  months  for  23  dollars  a 
month,  and  boards  himself.  He  pays  for  24  weeks' 
board  at  2  dollars  a  week.     He  expends,  besides,  for 

,a  hat,  3  dollars ;  for  boots,  4  dollars ;  and  for  other 
articles  of  clothing,  6  dollars.     How  much  does  he 
14*  L 


162  MISCELLANEOUS    EXAMPLES. 

lay  up,  deducting  the  above  expenses  from  the  amount 
of  his  earnings  ? 

12.  A  ship  sails  from  Boston  with  a  crew  of  19  men. 
At  New  Orleans,  3  of  the  crew  are  discharged,  and  7 
new  hands  taken  on  board.  The  ship  then  sails  to 
Liverpool,  where  5  of  the  crew  desert,  2  are  left  on  ac- 
count of  sickness,  and  3  new  hands  taken.  At  Havre, 
the  ship's  next  port,  4  men  desert,  and  3  new  hands 
are  taken,  when  the  ship  sails  for  Boston.  What  is 
the  number  of  her  crew  on  the  return  ?  and  what  do 
the  whole  wages  of  the  men  amount  to,  reckoning  the 
time  from  Boston  to  New  Orleans  one  month  ;  from 
New  Orleans  to  Liverpool,  one  month  and  a  half;  from 
Liverpool  to  Havre,  half  a  month  ;  from  Havre  home, 
one  month  and  a  quarter ;  allowing  9  dollars  a  month 
to  each  man  as  far  as  Liverpool,  and  8  dollars  a  month 
for  the  remainder  of  the  voyage  ? 

13.  What  is  the  sum  of  3  times  694  divided  by  2 ; 
9  times  1836  divided  by  27 ;  and  14  times  923  ? 

14.  What  is  the  amount  of  the  following  bill  of 
provisions,  namely,  18  barrels  of  beef,  at  6  dollars  a 
barrel ;  19  hundred  weight  of  ham,  at  8  dollars  a  hun- 
dred weight ;  173  barrels  of  flour,  at  5  dollars  a  barrel ; 
and  73  bushels  of  rye,  at  92  cents  a  bushel  ? 

15.  The  area  of  Pennsylvania  is  46,000  square  miles. 
If  the  population  is  1,740,000,  how  many  does  that 
give,  on  an  average,  to  each  square  mile  ?  How  many 
to  a  township  containing  30  square  miles  ? 

16.  New  York  has  the  same  area  as  Pennsylvania 
If  the  population  is  2,440,000,  how  many  is  that  to  a 
square  mile  ?     How  many  to  a  township  containing  30 
square  miles  ? 

17.  Virginia  has  an  area  of  64,000  square  miles. 
If  the  population  is  1,240,000,  how  many  are  there  to 
a  square  mile  ?  How  many,  on  an  average,  to  a  county 
containing  420  square  miles  ? 

18.  How  much  greater  was  the  increase  of  popula- 
tion in  Massachusetts,  (see  Section  II.)  from  1820  to 


REDUCTION.  163 

1830,  than  from  1810  to  1820  ?     How  much  greater 
from  1830  to  1840,  than  from  1820  to  1830  ? 

19.  The  area  of  Massachusetts  is  7500  square  miles. 
What  was  the  average  population  on  a  square  mile  in 
1810?     In  1820?     In  1830? 

20.  What  was  the  average  population  on  a  square 
mile,  in  the  state  of  New  York,  in  1810?  In  1820? 
In  1830  ? 

2J.!  Greenland  has  an  area  of  840,000  square  miles, 
and  20,000  inhabitants.  How  many  square  miles  does 
it  take  to  support  one  person  ?  How  many  would  live 
on  a  township  of  30  square  miles  ? 

22.  Siberia  contains  5,317,000  square  miles.  How 
many  states,  of  the  size  of  Massachusetts,  is  its  area 
equal  to?  It  contains  2,650,000  inhabitants.  How 
many  live,  on  an  average,  on  every  30  square  miles  ? 


SECTION    V. 

REDUCTION. 

The  object  in  reduction  is  to  change  a  quantity,  in 
one  denomination,  to  another,  which  shall  have  the 
same  value.  (See  Sec.  VI.,  Part  I.)  Higher  denomi- 
nations are  reduced  to  lower  by  multiplication. 

Examples. 

1.  Reduce  3  yards  to  feet. 

2.  Reduce  42  yards  to  feet. 

3.  Reduce  4  feet  to  inches. 

4.  Reduce  17  feet  to  inches. 

5.  Reduce  132  feet  to  inches. 
%                     6.  Reduce  16  yards  to  inches. 

7.  Reduce  21  yards  to  inches. 


1G4 


REDUCTION. 


8.  In  112  feet  7  inches  how  many  inches  ? 

9.  In  165  feet  4  inches  how  many  inches  ? 

10.  In  5  yards,  2  feet,  9  inches,  how  many  inches? 

11.  In  24  rods  how  many  feet  ? 

12.  In  87  rods  how  many  feet  ? 

13.  In  567  rods  how  many  inches  ? 

14.  In  7  rods  4  feet  how  many  feet  ? 

15.  In  31  rods,  2  feet,  6  inches,  how  many  inches? 

16.  In  131  £  how  many  shillings? 

17.  Reduce  781  £  to  shillings. 

18.  Reduce  758  £  to  shillings. 

19.  Reduce  19  shillings  to  pence. 

20.  Reduce  7£  11  shillings  to  pence. 

21.  Reduce  141  £  16  shillings,  4  pence,  to  pence. 

22.  Reduce  4£  7  shillings,  3  pence,  to  farthings. 

23.  Reduce  14  lbs.  8  oz.  avoirdupois,  to  oz. 

24.  Reduce  3  qrs.  9  lbs.  13  oz.  to  oz. 

25.  Reduce  44  cwt.  3  qrs.  19  lbs.  to  lbs. 

26.  Reduce  13  T.  12  cwt.  2  qrs.  to  lbs. 

27.  Reduce  3  lbs.  6  oz.  17  dwt.  Troy,  to  dwt. 

28.  Reduce  13  lbs.  2  oz.  14  dwt.  to  dwt. 

29.  Reduce  4  oz.  16  dwt.  to  grs. 

30.  Reduce  6  lb.  7  oz.  9  dwt.  4  grs.  to  grs. 

31.  Reduce  27  gallons  wine  measure  to  pints. 

32.  Reduce  7  hhds.  13  gals.  2  qts.  to  qts. 

33.  Reduce  1  hhd.  to  gills. 

34.  Reduce  174  bushels  to  quarts. 

35.  Reduce  73  bushels  to  pints. 

36.  Reduce  231  bushels  to  quarts. 

37.  In  13  square  feet  how  many  square  inches  ? 

38.  In  84  square  rods  how  many  square  feet  ? 

39.  Reduce  13  square  rods  to  inches. 

40.  Reduce  3  R.  17  rods  to  feet. 

41.  Reduce  5  A.  2  R.  14  rods  to  feet. 

42.  Reduce  17  solid  feet  to  inches. 

43.  Reduce  19  s.  yards  14  feet  to  inches. 

44.  Reduce  24  s.  yards,  8  feet,  504  inches,  to  inches. 


REDUCTION.  165 

45.  Reduce  6  cords  13  s.  feet  to  feet. 

46.  Reduce  27  cords  28  s.  feet  to  feet. 

47.  Reduce  45  cords  13  s.  feet  to  feet. 

48.  In  75  E.  e.  of  cloth,  how  many  quarters  ? 

49.  Reduce  78  yards  3  qrs.  to  quarters. 

50.  Reduce  194  yds.  1  qr.  to  nails. 

51.  Reduce  11  yds.,  3  qrs.,  2  nails,  to  inches. 

52.  Reduce  174  Fr.  e.  to  nails. 

53.  Reduce  4  m.,  5  fur.,  13  rods,  to  feet. 

54.  Reduce  17  m.,  6  fur.,  20  rods,  8  feet,  to  inches. 

55.  Reduce  21  £  17s.  3d.  to  pence. 

56.  Reduce  24  £  to  sixpences. 

57.  Reduce  95  £  3  s.  to  sixpences. 

58.  Reduce  45  £  5  s.  to  threepences. 

59.  Reduce  84  £  to  fourpences. 

60.  In  1  cwt.  3  qrs.  how  many  times  7  lbs.  ? 

61.  In -8  bis.  of  flour,  at  7  qrs.  each,  how  many  par- 
cels of  14  lb.  each  ? 

62.  In  13  bis.  of  cider,  at  31 J  gals,  each,  how  many 
times  8  gallons  ? 

63.  In  a  town  5  miles  wide  and  6  long  how  many 
acres  ? 

64.  How  many  acres  in  7500  square  miles  ? 


SECTION    VI. 

REDUCTION. 

Lower   denominations   are   reduced    to   higher   by 
division. 

Examples. 

1.  In  348  shillings  how  many  £  ? 

2.  Reduce  5000  shillings  to  £. 

3.  Reduce  13680  shillings  to  £. 

4.  Reduce  11040  pence  to  £. 


166  REDUCTION. 

5.  Reduce  11292  pence  to  £. 

6.  Reduce  20220  pence  to  £. 

7.  Reduce  1405  pence  to  £. 

8.  In  678  sixpences  how  many  £  ? 

9.  Reduce  549  threepences  to  £. 

10.  Reduce  974  threepences  to  shillings. 

11.  Reduce  1776  hours  to  days. 

12.  Reduce  13841  hours  to  days. 

13.  Reduce  1964210  minutes  to  days. 

14.  Reduce  3742196  seconds  to  hours. 

15.  Reduce  964  gills  to  gallons. 

16.  Reduce  84672  gills  to  gallons. 

17.  Reduce  6794  gallons  of  wine  to  barrels. 

18.  Reduce  3469  quarts  to  pecks. 

19.  Reduce  96431  pints  to  bushels. 

20.  Reduce  3846  qts.  to  bushels. 

21.  Reduce  5674  rods  to  furlongs. 

22.  Reduce  38961  rods  to  miles. 

23.  Reduce  76381  feet  to  rods, 

24.  Reduce  7960  inches  to  rods. 

25.  Reduce  7126734  inches  to  miles. 

26.  How  many  steps  of  2J-  feet  each  are  there  in  1 
mile  ? 

27.  A  man  walks  30  miles.     How  many  steps  does 
he  take,  2f  feet  each  ? 

28.  Reduce  179  lbs.  avoirdupois  to  cwt. 

29.  Reduce  413  lbs.  to  cwt. 

30.  Reduce  1048  oz.  to  quarters. 

31.  Reduce  4352  drams  to  lbs. 

32.  Reduce  6130  oz.  to  cwt. 

33.  Reduce  1280  dwt.  Troy  to  lbs. 

34.  Reduce  1511  dwt.  to  lbs. 

35.  Reduce  17812  grs.  to  lbs. 

36.  Reduce  720  square  inches  to  square  feet. 

37.  Reduce  1029  square  feet  to  yards. 

38.  Reduce  2203  square  inches  to  yards. 

39.  Reduce  3267  square  feet  to  rods. 


COMPOUND    ADDITION.  167 

40.  Reduce  5631  square  feet  to  rods. 

41.  Reduce  86  solid  feet  to  yards. 

42.  Reduce  191934  solid  inches  to  feet. 

43.  Reduce  2333  solid  inches  to  feet. 

44.  Reduce  876  solid  feet  to  yards. 

45.  Reduce  2293  solid  inches  to  yards. 

46.  In  92  qrs.  cloth,  how  many  E.  e.  ? 

47.  Reduce  361  nails  to  quarters. 

48.  Reduce  467  nails  to  yards. 

49.  Reduce  3741  inches  to  E.  e. 

50.  Reduce  467  yards  to  E.  e. 

51.  In  27  acres,  2  roods,  17  rods,  how  many  lots 
of  32  rods  each  ? 

52.  How  many  times  does  a  carriage  wheel  11£  feet 
in  circumference  go  round  in  one  mile  ? 

53.  In  35  tons  weight  how  many  wagon  loads  of 
22  cwt.  each  ? 

54.  In  186  £  12s.  how  many  guineas  of  21  shillings 
each  ? 

55.  In  75  yards  how  many  E.  e.  ? 

56.  How  many  cannon  balls,  at  24  lbs.  each,  will  it 
take  to  weigh  1  ton  gross  weight  ? 

57.  How  many  times  must  you  apply  a  pole  12  feet 
long  to  the  ground,  to  measure  1  mile  ? 

58.  How  many  10  gallon  kegs  may  be  filled  from 
17  hhds.  wine  measure  ? 


SECTION   VII. 

COMPOUND   ADDITION. 

When  numbers  are  used  without  being  applied  to 
any  particular  kind  of  quantity,  as  67,  84,  they  are 
called  abstract  numbers;   when  they  are  applied  to 


168 


COMPOUND    ADDITION. 


some  particular  quantity,  as  67  yards,  they  are  called 
denominate  numbers. 

When  several  numbers  of  different  denominations 
are  to  be  added,  as  3  £  7s.  +7 £  4s.,  it  is  called  Com- 
pound Addition. 

Examples. 

Operation. 

£     s.     d. 

3    14     9 

14    11     6 

18      6~ ¥   Ans. 


1.  3£14s.  9d.+14£lls.  6d. 


Set  down  numbers  of  the  same  denomination  under 
each  other.  Add  first  the  numbers  of  the  lowest  de- 
nomination. If  the  sum  amounts  to  more  than  one 
of  the  next  higher,  set  down  what  is  over,  and  carry 
the  number  of  the  higher  to  the  next  column.  So 
proceed  through  the  whole.  In  adding  the  last  col- 
umn, set  down  the  whole  amount. 

2.  5£  15s.  4d.+14£  17s*.  lld.+2£  6s.  5d. 

3.  13  £  14s.  6d.  2qr.  +  65£  17s.  lOd.  lqr. 

4.  48 £  16s.  +  73£  10s.  +  91£  15s.+16£  17s. 

5.  17s.  3d.+15s.  7d.  2qr.+14s.  Od.  2qrs. 

6.  Troy  Weight.  12  lbs.  1  oz.  16  dwt.  14  grs.  -f 
3  dwt.  17  grs. 

7.  17  lbs.  2  oz.  16  dwt.  5  gr.  +  6  lbs.  14  dwt.  17  grs. 

8.  21  lbs.  0  oz.  0  dwt.  3  grs.+19  dwt.  19  grs.  + 
13  dwt. 

9.  21  lbs.  3  oz.  16  dwt.  15  grs.  +  4  lbs.  4  oz.  17  dwt. 

13  grs. 

10.  Avoirdupois  Weight,  gross.    3  cwt.  0  qr.  17  lbs. 

14  oz.  +12  cwt.  3  qrs.  13  lbs.  12  oz. 

11.  IT.  14  cwt.  3  qrs.  17  lbs.  +  3  T.  17  cwt.  1  qr. 
21  lbs. 

12.  3  T.  16  cwt.  1  qr.  20  lbs.  6  oz.  +  5  cwt.  3  qrs. 
19  lbs.  13  oz. 


COMPOUND    ADDITION.  169 

13.  14  cwt.  1  qr.  20  lbs.  +18  cwt.  1  qr.  16  lbs.  + 
17  cwt.  1  qr.  11  lbs. 

14.  1  m.  3  fur.  17  r.  6  ft.  +  3  m.  5  far.  36  r.  12  ft. 

15.  65  m.  7  fur.  31  r.  +18  m.  19  fur.  23  r.  +19  m. 

4  fur.  17  r. 

16.  5  r.  15  ft.  +  27  r.  14  ft.  +16  r.  11  ft.  +  21  r.  12  ft. 

17.  15  r.  9  ft.  6  in.  +17  r.  3  ft.  4  in.  +25  r.  15  ft. 
11  in. 

*18.  3  fur.  17  r.  4  ft.  5  in.  +5  fur.  16  r.  14  ft.  9  in. 

19.  7  fur.  16  r.  3  ft.  2  in. +  6  fur.  34  r.  12  ft.  10  in. 

20.  13  m.  7  fur.  31  r.  +  6  m.  3  fur.  22  r.+ll  m. 

5  fur.  8  r. 

21.  Square  Measure.     2A.3R.6  p. +15  A.  1  R. 
17  p. 

22.  16  A.  2  R.  21  p.+8  A.  3  R.  33  p.  +  9  A.  2  R. 
9  P. 

23.  2  R.  15  p.  63  ft.  29  in.+l  R.  17  p.  31.  in.  + 
37  p.  18  inches. 

24.  13  p.  45  ft.  18  in. +19  p.  3  ft.  23  in. +17  p.  64 
ft.  71  inches. 

25.  Solid  Measure.     3  yds.  17  ft.  126  in. +  4  yds. 
23  ft.  64  in. 

26.  19  yards,  3  feet,  61  inches +  2  yards,  26  feet, 
1650  inches +  4  yards,  18  feet,  91  inches. 

27.  16  gals.  3  qts.  1  pt.  +  84  gals.  2  qts.  1  pt. 

28.  13  bushels,  3  pks.  4  qts. +  76  bushels,  3  pks.  5 
qts. 

29.  19  bu.  1  pk.  +  76  bu.  3  pks.  +  18  bu.  2  pks. 

30.  14  yds.  3  qrs.  1  n.  +21  yds.  2  qrs.  3  n. 

31.  3  days,  16  hours,  23  minutes, +17  days,  13  h. 
51m. 

32.  1  year,  11  weeks,  4  days, +  3  y.  14  w.  2  d. 

33.  7  deg.  14  min.  34  sec. +  19  deg.  20'  30". 

34.  21  deg.  7'  1 1"+14  deg.  18'  19". 

35.  61  deg.  26'  14" +  34  deg.  1'  8". 

15 


170  COMPOUND    SUBTRACTION. 

SECTION    VIII. 

COMPOUND   SUBTRACTION. 

Compound  subtraction  is  the  subtraction  of  num- 
bers of  different  denominations. 

Rule. — Set  numbers  of  the  same  denomination 
under  each  other.  Begin  at  the  right  hand,  setting 
down  the  remainder  found  by  subtraction,  under  its 
own  denomination.  If,  in  any  case,  the  minuend  is 
less  than  the  subtrahend,  borrow  one  from  the  next 
higher  denomination  of  the  minuend. 


Examples. 

1.  15£  8s.  9d.  — 11£  lis.  4d. 

2.  22£  19s.  8d.— 18£  15s.  9d. 

3.  13£4s.  6d.  — 9£  15s.  lOd. 


i 


Operation. 

£  s.  d. 
15  8  9 
11    11    4 


3    17    5  Ans. 

4.  18  bushels,  3    pecks,  4  quarts, — 16   bushels,  2 
pks.  5  qts. 

5.  44   bushels,   1    peck,  3    quarts, — 20   bushels,  2 
pks.  6  qts. 

6.  4  years,  3  months,*  14  days, — 2  years,  4  months, 
18  d. 

7.  28  years,  8  months,  5  days, — 19  years,  11  months, 
2  days. 

8.  18  gallons,  3  quarts,  1  pint, — 10  gallons,  1  quart, 
1  pint. 

9.  14  gallons  1  quart,  —  2  quarts  1  pint. 

10.  4  miles,  3  furlongs,  17  rods,  —  3  miles,  4  fur- 
longs, 21  rods. 

11.  19  miles,  7  furlongs,  11  rods,  —  9  miles,  6  fur- 
longs, 13  rods. 

*  Allow  30  days  to  a  month. 


COMPOUND    SUBTRACTION.  171 

12.  5  cwt.  3  quarters,  14  pounds,  —  4  cwt.  1  quarter, 
20  lbs. 

13.  12  cwt.  2  qrs.  21  lbs.,  — 9  cwt.  3  qrs.  23  lbs. 

14.  The  battle  of  Bunker  Hill  was  on  June  17, 
1775;  the  battle  of  Long  Hand,  August  27,  1776. 
What  was  the  length  of  time  between  them  ? 

•  - 15.  The  battle  of  the  Brandywine  was  September 
11,  1777.  How  long  was  that  after  the  battle  of  Long 
Island? 

16.  The  battle  of  Monmouth  was  June  28,  1778. 
How  long  was  that  after  the  battle  of  the  Brandy- 
wine  ? 

17.  The  army  of  Burgoyne  was  captured  October 
17,  1777;  that  of  Cornwallis,  October  9,  1781.  How 
long  between  these  events  ? 

18.  If  I  give  a  note  on  interest,  June  5,  1839,  and 
pay  it  March  10,  1841,  for  how  long  a  time  must  the 
interest  be  cast  ? 

19.  If  I  give  a  note  on  interest,  August  17,  1841, 
and  pay  it  June  9,  1843,  for  what  time  must  the  inter- 
est be  cast  ? 

20.  How  long  is  it  from  December  17,  1843,  to 
June  6,  1844? 

21.  How  long  from  September  9,  1842,  to  August 
3,  1844  ? 

22.  How  long  from  January  16,  1840,  to  July  17, 
1843? 

23.  How  long  from  November  14,  1841,  to  August 
21,  1844? 

24.  Boston  is  in  longitude  71°  M  W. ;  New  York, 
74°  1'.     What  is  the  difference  of  longitude  ? 

25.  Cincinnati  is  in  longitude  84°  277.  How  many 
degrees  W.  from  Boston  ? 

26.  How  many  degrees  of  longitude  is  Cincinnati 
west  from  New  York  ? 

•    27.-  How  many  degrees  of  longitude  is  Cincinnati 
west  from  Philadelphia,  whose  longitude  is  75°  ll7? 


172  COMPOUND    MULTIPLICATION. 

SECTION    IX. 

COMPOUND^IVIULTIPLICATION. 

Multiply,  first,  the  lowest  denomination.  If  the 
product  amounts  to  more  than  one  of  the  next  higher, 
set  down  what  is  over,  and  carry  the  number  of  the 
next  higher  to  the  next  product.  Multiply  the  next 
denomination  in  the  same  way,  and  so  on. 

Examples. 

Operation, 

1.  3T.  7cwt.  3qr.  multiplied  by  7. 


T.  cwt.  qr. 
3      7    3 

7 


2.  Multiply  4  £  5  s.  6  d.  by  2. 

3.  6£  4s.  3d.  1  qr.X3. 

4.  6  hours,  43  min.  15  sec.X4.         [23    14    1  Ans. 

5.  9  h.  11  m.  41  sec.X6. 

6.  14  days,  17  hours,  15  minutes,  3  seconds,  X  8. 

7.  8  bushels,  3  pecks,  1  quart,  X 16. 

8.  15  bushels,  2  pecks,  3  quarts,  1  pint,  X 12. 

9.  19  bushels,  1  peck,  2  quarts,  X  5. 

10.  1  mile,  5  furlongs,  13  rods,  12  feet,X2. 

11.  13  miles,  2  furlongs,  4  rods,  6  feet,  3  inches,  X 3. 

12.  2  cwt.  3  qrs.  16  lbs.X7. 

13.  14  cwt.  3  qrs.  14  lbs.  X  4. 

14.  What  is  the  weight  of  12  casks  of  lime,  each 
weighing  3  cwt.  1  qr.  17  lbs.  ? 

15.  How  many  yards  in  9  pieces  of  calico,  each 
measuring  23  yards  3  qrs.  ? 

16.  Troy  Weight.     6  lbs.  11  oz.  5  dwt.X7. 

17.  9  lbs.  8  oz.  16  dwt.  4  grs.X9. 

18.  Square  Measure.    7  acres,  2  roods,  17  rods,  X  9. 

19.  15  acres,  1  rood,  34  rods,Xl4. 


COMPOUND    DIVISION.  173 

SECTION    X. 

COMPOUND   DIVISION. 

Divide  the  highest  denomination  first,  and  set  the 
quotient  under  it.  Reduce  the  remainder,  if  any,  to 
the  next  lower  denomination  ;  add  it  to  those  of  the 
same  in  the  dividend,  and  divide  again.  And  so  on 
to  the  end. 


Examples. 


Operation. 

bu.  pk.  qt. 
2)7     3     5 


3     3    6|  Ans. 


1.  Divide  7  bu.  3  pks.  5qts.by2. 

2.  3£  lis.  6d.-^2. 

3.  18£  12s.  9d.^3. 

4.  26  hours  53  minutes  -f-  4. 

5.  Bo  hours,  10  minutes,  6  seconds,  -f-  6. 

6.  114  days,  18  hours,  0  minutes,  24  seconds, -r  8. 

7.  140  bushels  2  pecks  -r  16. 

8.  187  bushels,  1  peck,  2  quarts,  -f- 12. 

9.  Seven  men  are  entitled  to  equal  shares  of  67  £ 

13  s.  4d.     What  is  each  man's  share  ? 

10.  Three  men  are  to  receive  equal  shares  of  114£ 
19s.  9d.     What  is  each  man's  share? 

11.  What  is  one  fourth  of  13  lbs.  6  oz.  17  dvvt., 
Troy? 

12.  A  teamster  has  7  T.  11  cwt.  3  qrs.  of  mer- 
chandise, which  he  loads  on  three  wagons,  giving 
an  equal  load  to  each.     How  much  is  each  load? 

13.  If  you  divide  7  bushels  and  3  pecks  of  oats 
equally  among  5  horses,  how  much  will  each  receive? 

14.  If  a  piece  of  land,  containing  35  acres,  3  roods, 

14  rods,  be  divided  into  4  equal  parts,  how  much  will 
.each  part  be  ? 

15* 


174  MISCELLANEOUS    EXAMPLES. 


SECTION    XI 


MISCELLANEOUS    EXAMPLES. 

1.  A  teamster  loads  a  quantity  of  merchandise 
equally  on  three  wagons,  putting  on  each  1  T.  11 
cwt.  2  qrs.  Finding  these  loads  too  heavy,  he 
takes  a  fourth  wagon.  How  much  must  he  load 
on  each,  to  divide  the  whole  equally  among  the 
four  ? 

2.  A  man's  estate  amounts  to  784  £  10  s.  His 
wife  is  to  receive  214 £  15s.,  and  the  remainder  is 
to  be  divided  equally  among  4  children.  What  will 
be  each  child's  share? 

3.  Three  men  have  equal  shares  in  a  scaffold  of 
hay,  the  whole  of  which  weighs  5  T.  11  cwt.  What 
is  each  man's  share  ? 

For  the  following  examples  see  pp.  47,  48,  Part  I. 

4.  Rome  is  in  longitude  12°  28'  E.  from  London. 
What  time  is  it  at  Rome  when  it  is  noon  in  London  ? 

5.  Petersburg  is  in  longitude  29°  48'  E.  What 
time  is  it  at  Petersburg  when  it  is  noon  in  London  ? 

6.  Paris  is  in  longitude  2°  20'  E.  What  time  is  it 
at  Paris  when  it  is  noon  in  London  ? 

7.  Boston  is  in  longitude  71°  4'  W.  What  time  is 
it  in  Boston  when  it  is  noon  at  London  ? 

8.  New  York  is  in  longitude  74°  1'  W.  What 
time  is  it  in  New  York  when  it  is  noon  at  London  ? 

9.  What  time  is  it  in  Cincinnati,  84°  27'  W.}  when 
it  is  noon  in  Boston,  which  is  71°  4'  W.  ? 


DIVISIBILITY    OF    NUMBERS.  175 

SECTION    XII. 

DIVISIBILITY   OF  NUMBERS. 

In  order  to  ascertain  if  a  number  is  divisible  by 
either  of  the  following  numbers,  2,  3,  4,  5,  6,  8,  9,  10, 
or  any  combination  of  these,  see  Sec.  VIII.,  Part  I. 

To  ascertain  if  a  number  is  divisible  by  any  other 
number  than  the  above,  make  trial  of  other  prime 
divisors,  as  7,  11,  13,  17,  &c,  beginning  with  the 
smallest,  till  you  find  one  that  will  divide  the  given 
number,  or  find  that  it  is  indivisible. 

Remember,  that,  in  making  trial  by  these  numbers, 
you  need  not  go  higher  than  the  square  root  of  the 
given  number ;  for,  if  a  number  is  divisible,  one  of 
the  factors  will  certainly  be  as  small  as  the  square 
root.  Let  us  take  the  number  1079.  What  are  its 
prime  factors  ?  By  inspection  you  may  see  it  is  not 
divisible  by  2,  3,  5,  or  11,  consequently  not  by  4,  6,  8,  9, 
10,  or  12.  On  trying  it  by  7,  it  is  found  not  divisible 
by  7.  The  next  number  is  13.  This  divides  it,  giv- 
ing a  quotient,  83,  which  is  prime.  Its  only  factors, 
therefore,  are  13  and  83. 

Examples. 

1.  What  are  the  prime  factors  of  667? 

2.  What  are  the  prime  factors  of  406  ? 

#     3.    What  are  the  prime  factors  of  419?   Of  361?  Of 
742?    Of  281?    Of  316? 

4.  Prime  factors  of  941  ?  812?  749?  1116?  246? 
8104? 

5.  Prime  factors  of  266  ?    884  ?    1917  ?    376  ? 


176  REDUCTION    OF    FRACTIONS. 


SECTION    XIII. 

REDUCTION    OF    FRACTIONS. 

[See  Section  VIII.,  Part  I.] 

1.  Reduce  f-f  to  its  lowest  terms.     Ans.  £. 

2.  Reduce  f-f  to  its  lowest  terms. 

3.  Reduce  -$&  to  its  lowest  terms. 
4  Reduce  f §  to  its  lowest  terms. 

5.  Reduce  TVs  to  its  lowest  terms. 

6.  Reduce  £fi  to  its  lowest  terms.  In  this  exam- 
ple, it  is  not  evident  on  inspection  whether  the  two 
terms  of  the  fraction  have  any  common  divisor.  In 
such  cases,  you  may  adopt  the  following 

Mule  to  find  the   Greatest  Common  Divisor. 

Divide  the  greater  number  by  the  less ;  and  then 
take  the  divisor  for  a  new  dividend,  and  divide  it  by 
the  remainder ;  and  so  on,  till  there  is  no  remainder. 
The  last  divisor  will  be  the  greatest  common  divisor. 

Apply  the  above  rule  to  the  sixth  example. 


187)221(1 
187 

~34)187 
170 


17)34(2 
34 

00 


The  greatest  common  di- 
visor is,  therefore,  17 ;  and, 
dividing  the  terms  of  the 
fraction  by  this,  we  have 
for  the  lowest  terms,  i£. 


Demonstration  of  the  Rule. 

If  the  larger  number  is  a  multiple  of  the  smaller,  it 
is  evident  that  the  smaller  is  a  common  divisor  of  the^ 
two  numbers.     It  is  also  the  greatest  common  divisor ; 


THE    GREATEST    COMMON    DIVISOR.  177 

for  a  number  cannot  be  divided  by  any  number  greater 
than  itself.  The  answer,  therefore,  is  found  by  the 
first  division.  But  if  there  is  a  remainder,  next  find 
whether  the  remainder  will  exactly  divide  the  divisor. 
,  If  it  will,  it  will  divide  both  the  original  numbers  ; 
for,  if  it  will  divide  the  divisor,  it  will  divide  any 
multiple  of  the  divisor ;  and,  as  it  will  of  course 
divide  itself,  it  will  divide  any  multiple  of  the  divisor 
plus  itself.  Now,  the  larger  of  the  original  numbers 
is  a  certain  multiple  of  the  smaller  plus  the  remainder. 
If,  therefore,  after  the  first  division,  the  remainder  will 
divide  the  divisor,  it  is  a  common  divisor,  or  measure, 
of  the  two  numbers. 

It  is  also  the  greatest  common  divisor;  for,  as  it 
will  exactly  measure  the  smaller  of  the  two  numbers, 
it  will  exactly  measure  any  multiple  of  the  smaller. 
Now,  the  greater  number  is  a  certain  multiple  of  the 
smaller  plus  the  remainder.  The  remainder,  therefore, 
in  measuring  the  larger  number,  is  obliged  to  measure 
itself.  No  number  greater  than  itself  can  do  this. 
Therefore  the  remainder  is  the  greatest  common 
divisor.  If  the  work  has  to  be  carried  on  farther 
than  the  second  division,  the  same  reasoning  in  the 
demonstration  will  apply. 

Examples. 

7.  What  is  the  greatest  common  divisor  of  874 
and  437? 

8.  What  is  the  greatest  common  divisor  of  497 
and  451? 

\  .       9.    What  is  the  greatest  common   divisor  of  817 
and  913? 

10.  What  is  the  greatest  common  divisor  of  1007 
and  1219? 

11.  What  is  the  greatest  common  divisor  of  608 
,and  192? 

12.  What  is  the  greatest  common  divisor  of  869 

and  1343  ? 

M 


178  REDUCTION    OF    FRACTIONS. 

When  there  are  more  than  two  numbers,  first  find 
the  greatest  common  divisor  of  two  of  them,  and  then 
of  that  divisor  and  the  third  number. 

13.  What  is  the  greatest  common  divisor  of  608, 
941,  and  451  ? 

Whenever  it  is  possible,  by  inspection,  to  separate 
the  numbers  into  their  prime  factors,  this  method 
should  be  adopted. 

14.  What  is  the  greatest  common  divisor  of  94, 
804,  and  126  ? 

15.  What  is  the  greatest  common  divisor  of  1274, 
896,  and  580  ? 

Apply  the  above  rules  to  the  reduction  of  the  fol- 
lowing fractions :  — 

16.  Reduce  f£|  to  its  lowest  terms. 

17.  Reduce  to  their  lowest  terms,  f  If- ;  iff ;  Mil- 

18.  Reduce  to  their  lowest  terms,  f  f  g  ;  £ff  ;  f f £. 

19.  Reduce  to  their  lowest  terms,  TWg-?'  sWuJ  Iff- 

To  reduce  an  Improper  Fraction  to  a  Whole  or  Mixed 
Number. 

Perform  the  division  indicated  by  the  fraction  as  far 
as  possible.  If  there  is  a  remainder,  express  that  part 
of  the  division  by  placing  the  denominator  under  the 
remainder. 

20.  Reduce  T£  to  a  whole  or  mixed  number. 
Ans.   1TV 

21.  Reduce  SJ-  to  a  whole  or  mixed  number. 
Ans.  3f. 

22.  Reduce   to    a    whole    or    mixed   number,   -3ff9- ; 

if;  ff 

23.  Reduce   to    a   whole    or   mixed   number,  V>' 

if ;  W- 

24.  Reduce  the  improper  fractions,  -W9- ;  xf a ;  W  I 
if  a 


CHANGE    OF    NUMBERS    TO    HIGHER   TERMS.        179 


SECTION    XIV. 

CHANGE   OF   NUMBERS   AND   FRACTIONS  TO   HIGHER 
TERMS. 

It  is  sometimes  convenient  to  express  whole  numbers 
in  the  form  of  fractions,  and  to  express  fractions  in 
higher  terms  without  altering  the  value.  Thus  3  =  f, 
or-2^;   10  =  ^°-,  or-%0-. 

Examples. 

1.  In  4  how  many  fifths  ?     Ans.  20. 

2.  Express  the  value  of  4  in  fifths.     Ans.  2f- 

3.  Express  7  in  thirds. 

4.  Express  19  in  the  form  of  sevenths. 

5.  In  13  how  many  eighths  ? 

6.  Express  21  in  thirds. 

7.  Express  7  in  eighteenths. 

8.  Express  41  in  fourths. 

9.  In  3£  how  many  halves  ? 

10.  Change  4i  to  an  improper  fraction. 

11.  Change  17£  to  an  improper  fraction. 

12.  Change  24£  to  an  improper  fraction. 

13.  Change  to  an  improper  fraction,  18£ ;  112^- ; 
318^. 

14.  Change  f-  to  eighths,  without  altering  its  value. 

15.  Change  f  to  fifteenths. 

16.  Change  £  to  24ths ;  T3<y  to  80ths ;  f  to  99ths. 

17.  Change  T\  to  26ths  ;  T9¥  to  56ths ;  if  to  60ths. 

18.  Change  f  to  7ths. 

This  example  presents  a  difficulty,  because  the  re- 
quired denominator,  7,  is  not,  as  in  the  preceding  exam- 
ples, a  multiple  of  the  given  denominator,  4.  We  have 
seen,  however,  that,  if  we  multiply  or  divide  both  terms 
of  a  fraction  by  the  same  number,  the  value  will  not 
be  altered.     We  must,  then,  multiply  and  divide  both 


180       MULTIPLICATION    AND    DIVISION    OF    FRACTIONS. 

terms  by  such  numbers  as  will  give  us,  in  the  end,  7 
for  the  denominator.  The  question,  then,  is,  How 
can  we,  by  multiplication  and  division,  change  4  into 
7?  We  can  multiply  it  by  7,  which  will  give  28,  and 
then  divide  by  4,  giving  7  for  the  quotient.  Thus  the 
denominator  has  been  changed,  by  multiplication  and 
division,  from  4  to  7.  Now,  whatever  has  been  done 
to  the  denominator  must  be  done  to  the  numerator,  to 
preserve  the  value  of  the  fraction.  Multiplying  3  by 
7,  we  have  21 ;  dividing  this  by  4,  we  have  5£  for  the 

51 
required   numerator.      The  answer,  therefore,  is   — . 

This  fraction,  as  one  of  its  terms  contains  a  fraction  in 
itself,  is  called  a  Cornplex  Fraction. 

19.  Change  \  to  8ths ;  £•  to  9ths ;  T3T  to  7ths. 

20.  In    f   how   many    4ths  ?      How    many    5ths  ? 
6ths? 

21.  Change  4|  to  5ths  ;  84  to  llths ;  7£  to  4ths. 

22.  Change  22f  to  4ths  ;  I8i  to  7ths  ;  31^  to  5ths. 

23.  In  8£  how  many  3ds?  4ths?  5ths?  9ths? 

24.  In  19£  how  many  5ths  ?  4ths  ?  7ths  ? 

25.  In  9±  how  many  3ds?  5ths?  8ths? 

26.  In  13£  how  many  14ths  ?  15ths  ? 

27.  In  8£  how  many  17ths?  13ths? 

28.  In  20|  how  many  7ths  ?  8ths  ? 

29.  In  16i  how  many  4ths  ?  5ths  ? 

'  30.  In  ll£  how  many  37ths?  19ths? 


SECTION    XV. 

MULTIPLICATION   AND   DIVISION   OF   FRACTIONS. 

[See  Section  VUL,  Part  I.] 

1.  A  man  worked  72  days  for  |  of  a  dollar  a  day. 
What  did  his  wases  amount  to? 


MULTIPLICATION    AND    DIVISION    OF    FRACTIONS.        181 

2.  Multiply  £  by  46. 

3.  A  man  bought  139  bushels  of  apples  for  f  of  a 
dollar  a  bushel.     What  did  they  come  to  ? 

4.  Multiply  £  by  341 ;  fXl27. 

5.  A  garrison  of  700  soldiers  are  allowed  f  of  a 
pound  of  flour  a  day  for  each  man.  How  much 
would  they  consume  in  1  day  ?    How  much  in  7  days  ? 

6.  If  a  horse  eats  T\  of  a  bushel  of  oats  in  a  day, 
how  many  bushels  will  he  eat  in  365  days  ? 

7.  If  a  horse  eats  %£  cwt.  of  hay  in  a  week,  what 
part  of  a  cwt.  will  he  eat  in  one  day  ?  How  many 
cwt.  will  he  eat  in  a  year? 

8.  Three  men  gain,  by  an  adventure,  56£  dollars, 
which  they  are  to  share  equally.  .What  is  each  man's 
share  ? 

9.  What  is  |  of  187±  ?     What  is  *  of  91§  ? 

10.  T%X141?    T\X97?    |X140? 

11.  If  3£  cwt.  of  flour  be  divided  into  5  equal 
parts,  what  part  of  a  cwt.  will  each  share  be  ? 

12.  Divide  17J  by  8;  37£  by  14;  18+  by  9. 

13.  Divide  74±-=-5;  81l£-^-7;  38H-^-9. 

14.  A  man  and  his  son  agree  to  work  on  the  follow- 
ing terms  :  The  father  is  to  receive  |  of  a  dollar  a  day, 
and  the  son  f  as  much  as  the  father.  The  father 
works  18  days,  the  son  works  2£  times  as  long  as  the 
father.     What  do  the  wages  of  both  amount  to  ? 

15.  A  carpenter  agrees  to  work  on  a  house,  charg- 
ing If  dollar  a  day  for  himself,  and  f  as  much  for  his 
hired  man.  He  works  himself  38  days,  and  his  hired 
man  long  enough  to  have  his  wages  amount  to  7£  dol- 
lars more  than  his  own.  How  many  days  does  the 
hired  man  work? 

16.  A,  B,  and  O,  work  on  the  following  terms :  A 
is  to  receive  l£  dollar  a  day ;  B,  f  as  much  as  A ;  and 
C,  T7u  as  much  as  B.  A  works  13  days  ;  B,  twice  as 
many  days  as  A ;  and  C,  once  and  a  half  as  many  as 
B.     What  do  the  wages  of  the  three  amount  to  ? 

16 


JjK&      MULTIPLICATION    AND    DIVISION    OF    FRACTIONS. 

17.  Multiply  half  of  1672  by  J,  and  then  find  f  of 
the  product. 

18.  If  you  multiply  9f  by  643,  and  subtract  1572 
from  the  product,  what  will  TV  of  the  remainder  be  ? 

19.  18  times  72£  are  how  many  halves  of  16  times 
84i? 

20.  A  bought  one  third  of  a  quarter  ticket  in  a  lot- 
tery. The  price  of  a  whole  ticket  was  12  dollars. 
The  whole  ticket  drew  1000  dollars.  Of  the  prize 
money,  53<y  is  deducted  before  it  is  paid  over.  How 
much  did  A  receive  for  his  share  of  the  prize  ?  How 
much  did  he  gain,  after  deducting  the  cost  of  his 
ticket  ? 

21.  Encouraged  by  his  good  luck,  A  then  buys  18 
whole  tickets  in  the  same  lottery,  and  draws  in  all  25 
dollars,  subject  to  the  same  deduction  as  before. 
Does  he  gain  or  lose  by  his  whole  operation  in  the 
lottery,  and  how  much  ? 


SECTION    XVI. 

MULTIPLICATION   AND   DIVISION    OF  FRACTIONS. 

[See  Section  IX.,  Part  I.] 

1.  In  63  gallons,  how  many  bottles  of  T2^  of  a  gal- 
lon each  ? 

2.  From  7  lbs.^f  flour,  how  many  loaves  of  bread 
may  be  made,  each  containing  §  of  a  pound  of  flour  ? 

3.  Divide  U  +  i;  JH-»;  H4-J* 

4.  A  man  left  5846  dollars ;  §  of  the  whole  to  go  to 
his  wife,  and  the  remainder  to  be  equally  divided 
among  four  children.     What  was  each  child's  share  ? 

5.  From  a  piece  of  cloth  32  yards  long,  how  many 
coats  can  be  made,  each  requiring  2f-  yards  ? 


MULTIPLICATION    AND    DIVISION    OF    FRACTIONS.        183 

6.  From  a  stick  of  timber  26£  feet  long,  how  many 
blocks  can  be  cut,  each  T9^  of  a  foot  long  ? 

7.  If  a  family  consume  22£  pounds  of  flour  in  a 
week,  how  much  is  that  a  day  ?  How  much  will  they 
use  in  a  year  ? 

8.  Divide  134-^ ;  18} -Ml*;  86*-=-9£. 

9.  Multiply  251X16* ;  23*X2f;  31§X19£.  i 

A  fraction  of  a  fraction,  as  £  of  f ,  is  called  a  Com- 
pound Fraction.  This  is  reduced  to  a  simple  fraction 
by  multiplying  the  numerators  together  for  a  new 
numerator,  and  the  denominators  for  a  new  denomi- 
nator. It  is,  in  fact,  the  same  as  the  multiplication  of 
two  fractions  together. 

10.  What  is  i  of  §  of  76  ? 

11.  What  is  £  of  f  of  12?  What  is  f  of  f  of 
18}? 

12.  What  is  A  of  f  of  8*  ?  What  is  f  of  § 
of  34? 

13.  Divide  |  of  f  by  f     Divide  f  of  #  by  8£. 

14.  Multiply  i  of  37  by  19*.  Multiply  |  of  18 
by  19*. 

15.  What  is  the  value  of  32*  yards  of  cloth,  at  4£ 
dollars  a  yard  ? 

16.  What  do  17£  tons  of  hay  come,  to,  at  11*  dol- 
lars a  ton  ? 

17.  What  is  the  value  of  2l£  cords  of  wood,  at  4£ 
dollars  a  cord  ? 

18.  What  is  the  value  of  24£  barrels  of  apples,  at 
If  dollar  a  barrel  ? 

19.  What  is  the  amount  of  12£  shares  bank  stock, 
at  64 1£  dollars  a  share? 

20.  A  man  paid  away  £  of  his  money  for  a  horse, 
and  *  as  much  for  a  saddle,  3  dollars  for  a  bridle,  and 
had  1  dollar  left.  How  many  dollars  had  he  at 
first? 

21.  The  width  of  a  certain  room  is  f-  of  its  length. 
Its  height  is  }  as  great  as  its  width.     Its  door  is  6  feet 


184      ADDITION    AND    SUBTRACTION    OF    FRACTIONS. 

high,  which  is  £  of  the  height  of  the  room.     What  is 
its  length  ? 

22.  Of  a  certain  room,  the  width  is  f  of  the  length  ; 
the  height  is  £  of  the  width,  and  also  twice  and  one 
seventh  as  great  as  the  height  of  the  door,  which  is  7 
feet.     How  long  is  the  room  ? 

23.  Multiply  137|  by  31T4T.  Multiply  45f  by 
183f. 

24.  What  is  the  quotient  of  163f  divided  by  29T4F  ? 

25.  How  many  times  is  £f  of  57  contained  in  f  of 
1147*? 

26.  One  half  of  f  of  f  of  511  is  how  many  fifths 
of  19*? 

27.  Wliat  is  the  cost  of  31*  tons  of  hay  at  9f  dol- 
lars a  ton  ? 

28.  What  is  the  cost  of  109*  gallons  of  molasses  at 
29*  cts.  a  gallon  ? 

29.  What  is  the  cost  of  56*  cwt.  of  beef  at  4$  dol- 
lars per  cwt.  ? 

30.  *f  of  73  are  how  many  fifteenths  of  61*  ? 

31.  How  many  times  is  §  of  181*  contained  in 
1156* ? 


SECTION    XVII. 

ADDITION   AND   SUBTRACTION   OF   FRACTIONS. 

If  the  fractions  have  a  common  denominator,  per- 
form the  required  operation  on  the  numerators,  and 
place  the  result  over  the  common  denominator. 

1.  Add  A  +  A  +  f;   ^t  +  tV  +  t*;  tV  +  tV  +  tV 

2.  Subtract  f-f;  A~&J  «—  Uj  if— «• 

If  the  fractions  have  not  a  common  denominator, 
reduce  them  to  a  common  denominator,   (Sec.  IX., 


ADDITION    AND    SUBTRACTION    OF    FRACTIONS.        185 

Pt.  I.)  and  then  add   or  subtract,  as  the  question  re- 
quires. 

3.  Addi  +  i  +  f;    f  +  x4T  +  if;    i*6+f  +  i 

4.  Add  #  +  £  +  *£;    !  +  fr  +  i§;    £  +  t3t+iV 

5.  Subtract  5^  — f;    8T\  —  $;   f— A« 

;      6.    Add3H  +  57i  +  18i;    191  +  21*  +  3^. 

In  cases  like  the  above,  it  is  easiest  to  add  the 
whole  numbers  first. 

7.  Add  67TV  +  i  of  7i  +  i  of  22;  4T3T  +  *  of 
18f  +  |  of  13. 

8.  A  man  spent  for  various  articles  i  of  a  dollar, 
|  of  a  dollar,  ^  of  a  dollar,  T5^  of  a  dollar.  What 
part  of  a  dollar  did  he  spend  in  all? 

9.  From  56£  bushels  llf  bushels  were  taken. 
How  much  remained? 

10.  From  a  firkin  of  butter  containing  42£  lbs. 
18-J-i  lbs.   were  taken.     How  much  remained? 

11.  Add  1Ai  +  4f ;     Ui+gj    15*  +  7*  +  |t 

12.  Add  &tf+2*j    9i+?I;    13J  +  S+11*. 

5  11  7 

13.  Add  I9f+21«j    8f+g;    26i  +  «+g. 

14.  Add  4f  +  ^;  8i+^;   12i+3*+18A. 

15.  Subtract   19*  —  |g:    22£  — g;  18ft— 13A* 

lo  4o 

16.  Add  3i+8*  +  iy+A;    41£  +  ?*  +  — . 

40    Ui'        ¥^  7       28£ 
16* 


186  REDUCTION    OF    DENOMINATE    FRACTIONS. 


SECTION   XVIII. 

REDUCTION   OF    DENOMINATE   FRACTIONS. 

Denominate  fractions  are  fractions  of  numbers  when 
applied  to  a  particular  denomination.  See  Sec.  XL, 
Part  I. 

Examples. 

1.  What  part  of  a  bushel  is  £  of  a  quart  ?  f  of  a 
quart  ? 

Consider,  first,  what  part  of  a  bushel  one  whole 
quart  is. 

2.  What  part  of  2  bushels  is  i  of  a  quart  ?  f  of  a 
peck  ? 

3.  What  part  of  7  bushels  is  f 'of  a  peck  ?  £  of  a 
peck? 

4.  What  part  of  3  £  is  f  of  a  shilling  ?  T8T  of  a 
shilling  ? 

5.  What  part  of  a  shilling  is  i  of  a  penny  ?  -ft  of 
a  penny  ? 

6.  What  part  of  a  mile  is  £  of  a  rod  ?    f  of  a  rod  ? 

7.  What  part  of  3  furlongs  is  f  of  a  rod  ?  £  of  a 
rod? 

8.  What  part  of  a  mile  is  3  furlongs  19  rods  ? 

9.  What  part  of  a  mile  is  1  foot  ?    2  feet  ? 

10.  What  part  of  a  ton  is  6  cwt.  3  qrs.  7  lbs.  ? 

11.  What  part  of  a  ton  is  18  cwt.  1  qr.  6  lbs.  ? 

12.  What  part  of  a  square  rod  is  17  feet?  3l£ 
feet? 

13.  What  part  of  a  cord  is  12£  cubic  feet  ?  25£ 
cubic  feet  ? 

14.  What  part  of  a  cord  is  18J  cubic  feet  ?  34£ 
cubic  feet  ? 


REDUCTION    OF    DENOMINATE    FRACTIONS.  187 

15.  What  part  of  a  bushel  is  |  of  a  quart  ?  §  of  a 
quart  ? 

16.  What  part  of  a  bushel  is  7£  quarts  ?  1 1|  quarts  ? 

17.  What  part  of  a  week  is  4£  hours  ?    5T2T  hours  ? 

18.  What  part  of  a  week  is  7±  hours  ?    8£  hours  ? 

19.  What  part  of  3  hours  is  12  minutes  ?  15£ 
minutes? 

20.  What  is  the  value,  in  shillings  and  pence,  of  f 
of  a£? 

If  it  were  3  £,  there  would  be  60  shillings  ;  but  it  is 
not  3£,  but  one  seventh  of  that ;  therefore,  it  is  }  of 
60  shillings  =  8f  shillings.  To  find  the  value,  in 
pence,  of  f  of  a  shilling,  pursue  the  same  reasoning. 
If  it  was  4  shillings,  it  would  be  48  pence ;  but  it  is 
not  4  shillings,  but  j-  of  that  £=  6f  pence.  The  answer, 
then,  is  8  s.  6f  d. 

21.  What  is  the  value,  in  shillings  and  pence,  of  f- 
of  a£? 

22.  Value,  in  hours,  minutes,  and  seconds,  of  £  of 
a  week  ? 

23.  How  many  minutes  in  T5S  of  a  week  ? 

24.  How  many  minutes  and  seconds  in  A  of  an 
hour  ? 

25.  Value  of  f  of  a  gallon  ?  Value  of  ■£>  of  a  bar- 
rel of  wine? 

26.  How  many  oz.,  dwt.,  and  grs.,  in  t*  of  a  pound 
Troy? 

27.  What  is  the  value,  in  oz.,  dwt.,  and  grs.,  of  fV 
of  a  pound  Troy  ? 

28.  How  many  square  rods  in  f  of  an  acre  ? 

29.  How  many  square  feet  and  inches  in  T\  of  a 
square  yard? 


188  CHANGE    OF    DENOMINATE    INTEGERS. 


SECTION    XIX. 

CHANGE  OF  DENOMINATE  INTEGERS  TO  FRACTIONS. 

[See  Section  XL,  Part  L] 

Examples. 

1.  What  part  of  a  furlong  is  5  feet  ?    7i  feet  ? 
Consider,  first,  what  part  of  a  furlong  1  foot  is. 

2.  What  part  of  a  mile  is  17  feet  ?'   28^  feet  ? 

3.  What  part  of  a  mile  is  3l£  feet  ?    47f  feet  ? 

4.  What  part  of  a  ton  is  25J-  lbs.  ?    82£  lbs.  ? 

5.  What  part  of  a  ton  is  107i  lbs.  ?     130£  lbs.  ? 

6.  What  part  of  a  ton  is  3  qrs.  9   lbs.   8    oz.  ? 
17*  lbs.? 

7.  What  part  of  a  ton  is  4  lbs.  13  oz.  ?    19T%  lbs.  ? 

8.  What  part  of  an  acre  is  4  rods  17  feet  ?     139TV 
feet? 

9.  What  part  of  an  acre  is  113  rods  5l£  feet? 

10.  What  part  of  a  hogshead  of  wine  is  17  gals.  3 
qts.  1  pt.  ? 

11.  What  part  of  10  gallons  is  3£  pints  ?    8{-  pints? 

12.  What  part  of  a  guinea  is  3  s.  7d.  ?     14  s.  3£d.  ? 

13.  What  part  of  a  £  is  8s.  5£d.  ?     10s.  9f-d.  ? 

14.  What  part  of  a  week  is  3^  hours  ?    7TV  hours  ? 

15.  What  part  of  5  days  is  1  hour,  41  m.  15  sec.  ? 

16.  What  part  of  a  month   of  31    days   is   17   h. 
18J  m.  ? 

17.  What  part  of  an  ell  English  is  2  n.  1  in.  ? 

18.  What  part  of  21  yards  is  3£  qrs.?    9T3T  yards? 

19.  What  part  of  37£  yards  is  l£  ell  English  ? 

20.  What  part  of  7  cords  of  wood  is  2l£  cubic  feet  ? 

21.  What  part  of  31  £  is  5s.  3£d.  ?    14s.  2£d.  ? 


PRACTICAL    EXAMPLES.  189 


SECTION    XX. 


PRACTICAL  EXAMPLES. 


1.  What  is  the  cost  of  If  yds.  broadcloth  at  4 J  dols. 
a  yard  ?  $ 

2.  What  is  the  cost  of  2£  yds.  cloth  at  £  of  a  dollar 
a  yard  ? 

3.  What  is  the  value  of  12  bbls.  of  flour  at  4f  dols. 
a  barrel  ? 

4.  What  is  the  value  of  31  casks  of  lime  at  £  of  a 
dollar  a  cask  ? 

5.  What  is  5£  cwt.  of  beef  worth  at  4f  dols.  per 
hundred  weight? 

6.  What  is  the  cost  of  43£  bushels  corn  at  f-  of  a 
dollar  a  bushel  ? 

7.  If  a  horse  eat  l£  bushel  of  oats  in  a  week,  how 
much  will  he  eat  in  52  weeks  ? 

8.  What  will  be  the  cost  of  the  oats  for  52  weeks 
at  |  of  a  dollar  a  bushel  ? 

9.  What  part  of  a  cwt.  is  f  of  a  barrel  of  flour  con- 
taining 196  lbs.  ? 

10.  What  is  the  weight  of  f  of  a  lot  of  hay  weigh- 
ing 4T3<y  tons  ? 

11.  How  many  cubic  feet  are  there  in  f  of  £  of  a 
cord  of  wood  ? 

12.  A  man  sold  f  of  a  lot  of  wood,  the  whole  of 
which  was  17f  cords.     How  much  did  he  sell  ? 

13.  What  part  of  9  rods  in  length  is  10  feet  ? 

14.  What  part  of  a  square  rod  is  3  square  yards  ? 

15.  What  part  of  a  square  rod  is  5  square  feet  ? 

16.  What  is  36  square  feet  of  land  worth  at   9£ 
dollars  a* square  rod? 

17.  How  many  gallons  are  there  in  i£  of  a  barrel 
of  wine  ? 

18.  Two  men  bought  a  lot  of  hay  for  ll£  dollars. 


190  DECIMAL    FRACTIONS. 

One  took  13  cwt. ;  the  other,  the  remainder,  which 
was  8£  cwt.     What  ought  each  to  pay  ? 

19.  Two  men  divided  a  lot  of  wood,  which  they 
purchased  together  for  27£  dollars.  One  took  5£ 
cords  ;  the  other,  8  cords.     What  ought  each  to  pay  ? 

20.  The  main-spring  of  a  watch  weighs  about  1 
dwt.  12  grs.  Troy  weight.  Estimating  its  worth  at  £ 
of  a  dollar,  what  would  a  pound  Troy  of  steel  be 
worth,  after  it  was  manufactured  into  watch  main- 
springs, allowing  nothing  for  waste  in  manufacturing? 

21.  A  hair-spring  of  a  watch  weighs  j-  of  a  grain 
Troy.  Estimating  its  value  at  3  cents,  what  would 
be  the  value  of  1  lb.  Troy  of  steel,  made  into  hair- 
springs, allowing  nothing  for  waste  ? 

22.  Two  men  hired  a  horse  one  week  for  6£  dol- 
lars. One  rode  him  70  miles ;  the  other,  84.  How 
much  ought  each  to  pay  ? 

23.  A  stack  of  hay  is  bought  by  two  men  for  76£ 
dollars,  to  be  paid  for  in  proportion  to  the  amount  of 
hay  each  one  takes.  One  takes  3£  tons ;  the  other, 
the  remainder,  which  was  2|  tons.  How  much  ought 
each  to  pay  ? 


SECTION    XXI. 

DECIMAL  FRACTIONS. 

Addition  and  Subtraction.     Sec.  XII.,  Pt.  I. 

Rule.  —  Set  down  figures  of  the  same  order  undei 
each  other,  units  under  units,  tenths  under  tenths,  &c* 
The  figures  in  the  answer  will  be,  each,  of  the  same 
order  with  the  perpendicular  line  of  figures  over  it. 
With  this  in  mind,  insert  the  point  in  its  proper  place. 


DECIMAL    FRACTIONS.  191 

Examples. 

1.  24.5  +  68.3+17.14  +  87.96  +  3.125. 

2.  165.3  +  96.45  +  8.43 1+  .641+  9412.5. 

3.  450.61+27.134  +  89.4216  +  . 984. 

4.  64.25  +  3.125  +  87.25+181.7. 

5.  125. 17+ 34.27+.  125 +  3761.5. 

6.  186.4  —  27.31;  800.4  —  21.67. 

7.  34.21—18.525;  94.31—81.167. 

8.  167.51—35.125;  204.5  —  31.09. 

9.  20.41—3.817;  601.4—517.24. 

10.  648.62  — .541;  346.4  —  91.324. 

11.  5.1—1.324;  .5  — .0067. 

12.  .81— .126;  .94  — .3816. 

Multiplication  and  Division. 

Rule.  —  Perform  the  operation  as  in  whole  num- 
bers ;  and,  in  inserting  the  point,  remember  that  every 
product  or  dividend  has  as  many  decimal  places  as 
both  the  factors  which  produce  it. 

13.  124.3X87;  321.67X24.3. 

14.  97.125X6;  31.4X.125. 

15.  37.5X.94;   18.4X64. 

16.  21X.106;  312X.05. 

17.  31. IX. 004;  18.61x.03. 

18.  641X.41;  843.5X.95. 

19.  184.2X.121;  35.6X.025. 

20.  .625X71;  .875X31.5. 

21.  84-4-. 012;  965-K 15. 

22.  1.65 -M5;  846 -h 3.4. 

23.  1640 -K 96;  425-^.055. 

24.  1-K001;  2 -4-. 0002. 

25.  .001H-2;  384-K0012. 

26.  96-K024;  64-^.016. 

27.  1827^.9;  34-f-.17. 


192  REDUCTION    OF    FRACTIONS    TO    DECIMALS. 

28.  .634-8;  .154-14. 

29.  .484-. 9;  .334-16. 

30.  1814-.41;  414-.6. 

31.  354- .36;  484-.47. 

32.  .174-31;  .264-.013. 

33.  43 4-. 06;  45  4-. 003. 

34.  75 4-.  125;  .954-. 04. 

35.  .18  4-. 0045;  114- .34. 

36.  9  4-. 0225;  .74-. 035. 

37.  804-.18;  514- .031. 

38.  .55X.031;  71.4X.13. 

39.  8.44-.021;  .65-T-.8. 

40.  1.21X.09;  .14X.03. 

41.  .64X.31;  .08X.009. 

42.  364-. 13;  284-11.4. 

43.  40.1^-8;  644-.9. 

44.  81.4 4- .03;  74-. 4. 

45.  94- .5;  154-.7. 

46.  80.2X.03;  164-.9. 

47.  105.44-37.15. 

48.  118.75  4-. 0044. 


SECTION    XXII. 

REDUCTION  OF  VULGAR  FRACTIONS  TO  DECIMALS. 

[See  Section  XIII.,  Part  I.] 

Examples. 

1.  Reduce  f  to  a  decimal. 

Rule.  —  Reduce  the  numerator  to  tenths,  and  divide 
the  result  by  the  denominator,  pointing  off  as  required 


REDUCTION  OF  FRACTIONS  TO  DECIMALS.     193 

in  division  of  decimals.     If  there  is  a  remainder,  reduce 
it  to  the  next  lower  order,  and  divide  again. 

2.  .Reduce  f  to  a  decimal. 

3.  Reduce  to  decimals,  £  j  £. 

4.  Reduce  to  decimals,  TV  >  A  J  tV 

5.  Reduce  to  decimals,  T\ ;  ft ;  if. 

If  the  fractions  are  reducible  to  decimals  without  a 
remainder,  obtain  the  answer  exactly.  If  they  are 
irreducible,  obtain  the  proximate  answer  to  four  places, 
and  annex  the  fractional  remainder.  In  order  to  know 
if  a  fraction  is  exactly  expressible  in  decimals,  see  Sec- 
tion XIII. ,  Part  I.,  as  directed  above. 

6.  Reduce  to  decimals,  £f  ;  |J ;  § J. 

7.  Reduce  to  decimals,  ^T ;  u9T ;  rfc. 

8.  Reduce  to  decimals,  T£ ;  y^V;  ttt- 

9.  Reduce  to  decimals,  ¥Vt  J  Ht  J  M- 

10.  Reduce  to  decimals,  T^ ;  Tf F ;  ^f  T. 

In  ordinary  transactions,  it  is  usual  to  carry  the  deci- 
mal answer  to  three  or  four  places.  The  remainder  is 
then  so  small  in  value,  that  it  may  be  dropped  as  of  no 
importance.  At  whatever  place  you  stop,  however, 
the  decimal  obtained,  and  the  fractional  remainder, 
when  added  together,  will  exactly  equal  the  original 
fraction. 

11.  In  order  to  show  this,  we  will  take  }.     Redu- 

7)10  ■      ;. 

eing  it,        —         we  obtain,  at  the  first  step,  1  tenth 

I~TT> 

-H  of  1  tenth.    Adding  these,  ^  +  T\  —  fg-  == f,  which 
is  the  original  fraction. 

We  now  carry  the  reduction  one  step  farther: 
7)100 

—  ,  2  We  obtain  14  hundredths  +  ?  of  a  hun- 
dredth. Adding  these,  tWf+t&tf  =  +&{*  =  *,  the  original 
fraction. 

17  „ 


194  REDUCTION    OF    DENOMINATE    INTEGERS. 

We  will  carry  the  redaction  one  step  farther ; 
7)1000 

— —  We  obtain  142  thousandths  +  1   of  a 

142  +  f.  ^T   . 

thousandth.  Adding  these,  by  using  the  common 
denominator  7000,  AVs  +  *A*  =  $ U%  =  h  the  original 
fraction. 

12.  Reduce  T2T  to  a  decimal  of  one  figure,  with  the 
remainder;  carried  to  2  places,  with  the  remainder; 
carried  to  3  places,  with  the  remainder. 

13.  Reduce  T4F  to  a  decimal  of  7  places. 

14.  Reduce  T73  to  a  decimal  of  9  places. 

15.  Reduce  F8T  to  a  decimal  of  10  places. 

Repeating  and  Circulating  Decimals. 

When  a  fraction  is  irreducible,  the  decimal  figure 
will  either  repeat,  as,  £  =  .333  +  ;  or  the  decimal  fig- 
ures obtained  by  the  partial  reduction  will,  after  a  time, 
recur  again,  in  the  same  order  as  at  first.  Thus,  T\ 
gives  .090909  +,  and  so  on,  without  end.  When  the 
same  figure  is  repeated  continually,  it  is  called  a  repeat- 
ing decimal ;  when  the  same  series  of  different  figures 
recurs,  it  is  called  a  circulating  decimal. 


SECTION    XXIII. 

REDUCTION   OF   DENOMINATE   INTEGERS   TO 
DECIMALS. 

1.  Reduce  5s.  lid.  to  the  decimal  of  a  £. 

First,  reduce  the  quantity  to  the  vulgar  fraction  of 
a  £.    Then  reduce  that  vulgar  fraction  to  a  decimal. 

2.  Reduce  3s.  2£d.  to  the  decimal  of  a  £. 


INTEGRAL    VALUE    OF    DENOMINATE    DECIMALS.      195 

3.  Reduce  5d.  to  the  decimal  of  a  guinea. 

4.  Reduce  3  qts.  to  the  decimal  of  a  bushel. 

5.  Reduce  2£  pints  to  the  decimal  of  a  gallon. 

6.  Reduce  3  feet  5  inches  to  the  decimal  of  a  rod. 

7.  Reduce  7  feet  8  inches  to  the  decimal  of  a  rod. 

8.  Reduce  15  rods  9£  feet  to  the  decimal  of  a  fur- 
long. 

9.  Reduce  23  rods  13  feet  to  the  decimal  of  a  mile. 

10.  Reduce  5  hours  18  minutes  to  the  decimal  of 
a  day. 

11.  Reduce  21  hours  6  minutes  to  the  decimal  of  a 
week. 

12.  Reduce  12£  square  rods  to  the  decimal  of  an 
acre. 


SECTION    XXIV. 

TO   FIND   THE   INTEGRAL  VALUE   OF  DENOMINATE 
DECIMALS. 

1.  What  is  the  value  of  .7  of  a  rod? 

Supposing  the  quantity  was  7  rods,  its  value  in  feet 
would  be  found  by  multiplying  it  by  16£;  16£X7  = 
115£,  or  115.5.  But  it  was  ftot  7  rods,  but  7  tenths  of 
a  rod,  whose  value  we  wish  to  find.  The  answer  ob- 
tained, therefore,  is  10  times  too  large.  Dividing  by 
10,  it  is  11.55=11  feet  and  55  hundredths.  In  order 
to  find  the  value  in  inches  of  55  hundredths  of  a  foot, 
we  will  call  it  55  feet;  the  answer  is,  55X12  =  660  = 
660  feet.  But,  as  we  regarded  the  55  as  100  times 
greater  in  value  than  it  is,  the  answer  is  100  times  too 
large.  Dividing  it  by  100,  the  answer  is  6.60  inches, 
=  6  inches  and  60  hundredths  or  6  tenths. 

The  above  analysis  shows  the  nature  of  the  opera- 
tion in  all  cases. 


196  PRACTICAL  EXAMPLES. 

2.  What  is  the  value,  in  feet  and  inches,  of  .3  of  a 
rod  ? 

3.  What  is  the  value  of  .94  of  a  rod? 

4.  What  is  the  value  of  .26  of  a  rod? 

5.  How  many  shillings  and  pence  are  there  in  .65 
of  a£? 

6.  How  many  shillings  and  pence  are  there  in  .8 
of  a£? 

7.  How  many  pence  are  there  in  .7  of  a  shilling  ? 

8.  How  many  pence  are  there  in  .16  of  a  shilling? 

9.  What  is  the  value  of  .19  of  a  £  ? 

10.  What  is  the  value  of  .74  of  a  bushel  ? 

11.  What  is  the  value  of  .9  of  a  bushel  ? 

12.  What  is  the  value,  in  rods  and  feet,  of  .7  of  an 
acre  ? 

13.  What  is  the  value  of  .9  of  an  acre  ?    . 

14.  What  is  the  value  of  .12  of  an  hour  ? 

15.  How  many  minutes  and  seconds  in  .15  of  an 
hour  ? 

16.  Find  the  value  of  .34  of  a  week. 

17.  Find  the  value  of  .162  of  a  week. 

18.  Find  the  value  of  .84  of  a  minute. 

19.  How  many  feet  in  .761  of  a  cord? 

20.  How  many  feet  and  inches  in  .2  of  a  cord  ? 

21.  How  many  feet  in  .74  of  a  cord  ? 

22.  How  many  feet  in  .13  of  a  cord? 


SECTION    XXV. 

PRACTICAL  EXAMPLES. 


1.  Add  $1.50  +  $.375  +  $.0625  +  $.1875  +  $5.00. 

2.  Add  $34.75 +  $6.00  +  $.375 +  $.08. 

3.  A  man  had  $50,  and  spent  $.375  of  it.     How 
much  had  he  left  ? 


DECIMAL    FRACTIONS.  197 

4.  A  man  had  $10.00,  and  spent  $.875  of  it.    How 
much  had  he  left  ? 

5.  A   watch   cost    $45,675;    the   chain   and   key, 
$4,845.     What  did  the  whole  cost  ? 

6.  The  owner  then  sold  the  watch,  chain,  and  key, 
for  $48,375.     How  much  did  he  lose? 

7.  A  man  set  out  on  a  journey  with  $10.00.     The 
first  day,  he  spent  $1,125.     How  much  had  he  left? 

8.  The  second  day,  he  spent  $1,425.     How  much 
had  he  left? 

9.  The  third  day,  he  spent   $1.67.      How  much 
had  he  left  ? 

10.  The  fourth  day,  he  spent  $.875.     How  much 
had  he  left  ? 

11.  What  is  the  cost  of  21  11*5.  of  flour  at  $.05 
per  pound? 

Why  do  you  point  off  two  decimals  in  the  answer  ? 

12.  What  is  the  cost  of  35  lbs.  of  flour  at  $.045 
per  pound  ? 

Why  do  you  point  off  three  decimals? 

13.  What  is  the  cost  of  12.5  lbs.  of  flour  at  $.05 
a  pound  ? 

14.  What  is  the  cost  of  15.5  lbs.  of  flour  at  $.045 
a  pound  ? 

15.  What  is  the  cost  of  26.25  lbs.  of  flour  at  $.0375 
a  pound? 

16.  What  is  the  cost  of  13.75  lbs.  of  flour  at  $.0425 
a  pound  ? 

17.  What  is  the  cost  of  15  barrels  of  flour  at  $4.75 
a  barrel  ? 

18.  What  is  the    cost   of  17.5  barrels   of  flour  at 
$5.25  a  barrel? 

19.  What  is  the  cost  of  3  tons  of  hay  at  $7.56  a  ton  ? 

20.  What  is  the  cost  of  13.5  tons  of  hay  at   $9.00 
a  ton? 

21.  What  are  17  barrels  of  cider  worth  at   $1.75 
a  barrel  ? 

17* 


198  PRACTICAL  QUESTIONS. 

22.  What  cost  16  gallons  of  molasses  at  $.345  a 

gallon  ? 

23.  Divide  $1.05  into  21  equal  parts.     What  will 
each  part  be  ? 

24.  How  many  pounds  of  flour  will  $15.75  buy,  at 
$.045  a  pound? 

25.  How  many  times  is  $.05  contained  in   $.625? 

26.  How  many  pounds  of  flour  can  be  bought  for 
$.6975  at    $.045   per  pound? 

27.  How    many    times    is     $.0375    contained    in 
$984,375? 

28.  How   many    times    is     $.0425    contained    in 
$584,375? 

29.  How  many  barrels  of  flour  will   $71.25   buy, 
at   $4.75  per  barrel? 

30.  How  many  barrels  of  flour  will  $91,875  buy, 
at   $5.25  per  barrel? 

31.  How  many  tons   of  hay   can   be   bought  for 
$22.68,  at   $7.56  per  ton? 

32.  How    many    times    is     $9.00    contained    in 
$121.50? 

33.  A  shipmaster  paid    $29.75  for  ballast,  giving 
$1.75  a  ton.     How  many  tons  did  he  buy? 

34.  How  many  times  is  $.345  contained  in  $5.52? 

35.  What  cost  14  lbs.  of  flour  at  $.045  a  pound, 
and  28  lbs.  of  sugar  at   $.095  a  pound? 


SECTION    XXVI. 

PRACTICAL  QUESTIONS  IN  VULGAR  AND  DECIMAL 
FRACTIONS. 

1.  Bought  7  cwt.  15  lbs.  sugar  at  $6.62£  per  cwt., 
and  sold  it  at  7  cents  per  pound.    What  was  the  gain  ? 


VULGAR  AND  DECIMAL  FRACTIONS.         199 

2.  Bought  156  gallons  of  wine  at  93  cents  per  gal- 
lon, and  sold  it  at  34  cents  per  quart.  What  was  the 
gain  ? 

3.  Bought  7  cwt.  1  qr.  11  lbs.  coffee  at  $12.50  per 
cwt.,  and  sold  it  at  14  cents  per  pound.     What  gain? 

4.  Bought  37  yards  broadcloth  at  $5.25  per  yard. 
Sold  20  yards  of  it  at  $7.00  per  yard,  and  the  remain- 
der at  $6.31  per  yard.     What  was  the  gaiu  ? 

5.  Bought  24  yards  broadcloth  at  $6.40  per  yard. 
Sold  22£  yards  at  $7.25  per  yard,  and  the  remnant  for 
5  dollars.     What  was  the  gain? 

6.  Bought  87  E.  e.  calico  at  17  cents  per  E.  e.,  and 
sold  it  at  21  cents  per  yard.     What  gain  ? 

7.  Bought  4  dozen  books  at  $1.50  per  dozen,  and 
sold  them  at  16  cents  each.     What  gain  ? 

8.  Bought  13  dozen  brooms  at  $1.04  per  dozen, 
and  sold  them  at  15  cents  each.     What  gain? 

9.  Bought  5£  dozen  mats  at  $3.40  per  dozen,  and 
sold  them  at  36  cents  each.     What  gain  ? 

10.  Bought  17  bushels  of  salt  at  65  cents  per  bushel, 
and  sold  it  at  21  cents  per  peck.     What  gain  ? 

11.  Bought  one  barrel  of  wine  at  78  cents  per  gal- 
lon, and  sold  it  at  16  cents  per  pint.     What  gain? 

12.  Bought  3  dozen  baskets  at  $2.05  per  dozen, 
and  sold  1  dozen  at  31  cents,  1  dozen  at  37  cents,  and 
1  dozen  at  42  cents  each.     What  gain? 

13.  Bought  48  yards  broadcloth  at  $5.62  per  yard. 
Lost  17  yards  by  fire,  and  sold  the  remainder  at  $6.25 
per  yard.     How  much  gain  or  loss  ? 

14.  Bought  a  hogshead  molasses,  containing  131 
gallons,  at  34  cents  per  gallon.  16  gallons  leaked  out. 
Sold  the  remainder  at  37  cents  per  gallon.  What  gain 
or  loss  ? 

15.  Bought  3£  dozen  axes  at  $6.80  per  dozen,  and 
sold  them  at  92  cents  each.     What  gain  ? 

16.  Bought  7  dozen  pails  at  $1.42  per  dozen,  and 
sold  them  at  21  cents  each.     What  gain  ? 


200  REDUCTION    OF    CURRENCIES. 

17.  Bought  8£  dozen  shovels  at  $9.25  per  dozen, 
and  sold  them  at  $1.00  each.     What  gain? 

18.  Bought  74  yards  carpeting  at  73  cents  per  yard, 
and  sold  it  at  87£  cents  per  yard.     What  gain  ? 

19.  Bought  164  bushels  corn  at  54  cents  per  bushel. 
Sold  93  bushels  at  67  cents,  and  the  remainder  at  50 
cents,  per  bushel.     How  much  loss  or  gain? 

20.  Bought  75  barrels  apples  at  $1.37  per  barrel. 
Lost  15  barrels  by  decay,  and  sold  what  remained  at 
$2.12  per  barrel.     What  loss  or  gain? 

21.  Bought  13  dozen  oranges  at  7  cents  per  dozen. 
Lost  by  decay  2£  dozen,  and  sold  the  remainder  at  2£ 
cents  each.     What  gain  ? 

22.  Bought  15  dozen  pairs  of  shoes  at  $4.87  per 
dozen,  and  sold  them  at  63  cts.  per  pair.     What  gain  ? 

23.  Bought  18£  thousand  of  boards  at  $9.50  per 
thousand.  Sold  6  thousand  at  $12.25  per  thousand, 
and  the  remainder  at  $8.42  per  thousand.    What  gain? 

24.  Bought  21£  cords  wood  at  $4.75  per  cord. 
Sold  8  cords  at  $5.50  per  cord,  and  the  remainder  at 
$4.25  per  cord.     What  gain  or  loss? 

25.  Bought  209  bushels  apples  at  27  cents  per 
bushel.  Sold  46  bushels  at  49  cents  per  bushel,  and 
the  remainder  at  25  cents  per  bushel.     What  gain  ? 


SECTION     XXYII. 

REDUCTION   OF   CURRENCIES. 
English  Currency. 

1.  Reduce  67  £  to  dollars  and  cents. 
As  4s.  6d.  or  54 d.  =  $1.00,  (see  Table,  p.  40,)  and 
20  s.  or  240  d,  =  1  £,  1  dollar  is  ^fc  of  a  £.    Reducing 


REDUCTION  OF  CURRENCIES.  201 

this  fraction  to  its  lowest  terms,  it  is  T9D.  The  ques- 
tion, therefore,  is  this:  In  67  £  how  many  -^  of  a  £? 
Dividing  67  by  the  fraction,  we  have  297£  dollars 
for  the  answer.  The  fraction  £  gives  77  cents  and 
7  mills. 

2.  Reduce  87  £  to  dollars  and  cents. 

3.  Reduce  104  £  to  dollars  and  cents. 

4.  Reduce  64  £  to  dollars  and  cents. 

5.  Reduce  167  £  to  dollars  and  cents. 

6.  Reduce  520  £  to  dollars  and  cents. 

7.  Reduce  84 £  6  s.  to  dollars  and  cents. 

First  reduce  the  6  s.  to  the  decimal  of  a  £  j  ^  =  .3. 
The  sum  then  is,  84.3 £.  Reduce  it  in  the  same  way 
as  the  cases  above. 

8.  Reduce  124  £   13  s.  to  Federal  money. 

9.  Reduce  36  £  9  s.  6  d.  to  Federal  money. 

10.  Reduce  71  £   18s.  4d.  to  Federal  money. 

To  Reduce  Federal  Money  to  Sterling. 

11.  In  684  dollars  how  many  pounds,  shillings,  and 
pence  ? 

As  *  1.00  =  A  of  a  £,  1£=-4/  of  $1.00.  The 
question  therefore  is,  In  684  dollars  how  many  4^°-  of 
$1.00?  Dividing  by  the  fraction,  we  have  for  the 
answer,  £153.9,  or  153£   18s. 

12.  In  $74.25  how  many  pounds,  shillings,  and 
pence  ? 

13.  Reduce   $186.40  to  Sterling  money. 

14.  Reduce   $564.35  to  Sterling  money. 

15.  Reduce   $640.15  to  Sterling  money. 

The  comparative  value  of  the  dollar  and  the  pound 
sterling,  as  given  above,  is  called  the  nominal  par 
value.  The  actual  value  of  the  pound  is  higher  than 
is  here  given.     This  difference  is  usually  estimated  in 


202  REDUCTION  OF  CURRENCIES. 

trade  by  adopting  the  nominal  par  value,  given  above, 
as  the  basis  of  the  calculation,  and  then  adding  or 
subtracting  a  certain  per  cent.,  as  8  or  10  per  cent.,  to 
compensate  for  the  inequality  of  value. 

Canada  Currency. 
5s.  =  60d.=  *1.00. 

16.  In  74  £  15  s.  how  many  dollars  and 'cents? 

As  *  1.00  =  60 d.,  and  l£=240d.  $1.00  is  Jft,  or 
I  of  a  pound ;  multiplying  by  4,  the  answer  is 
$299.00. 

17.  In  £  126  12  s.  Canada  currency  how  many  dol- 
lars and  cents  ? 

18.  Reduce  $841.50  to  Canada  currency. 

/  New  England  Currency. 

6s.=72d.  =  $1.00. 

19.  In  64 £  8s.  how  many  dollars  and  cents? 
$1.00  =  ^=  T\  of  a  £.     Reduce    the   8s.  to  a 

decimal  of  a  pound,  and  divide  by  the  fraction ;  we 
have  $214.66f. 

20.  Reduce  120  £  12  s.  6d.  to  Federal  money, 

New    York  Currency. 
8s.  =  96d.=  $1.00. 

21.  Reduce  146  £  6  s.  4d.  to  Federal  money. 

As  $1.00  =  2Vff  =  fV  of  a  pound,  reducing  the  shil- 
lings and  pence  to  the  decimal  of  a  pound,  and  divid- 
ing by  the  fraction,  we  have  $365.75. 

22.  Reduce  54  £  10  s.  6d.  to  Federal  money. 

Pennsylvania  Currency. 
7s.  6d.  =  90d.=  *1.00. 

23.  Reduce  16  £  5  s.  6d.  to  Federal  money. 
$1.00  =  29^=f  of  a  £. 

24.  Reduce  7£  8  s.  9d.  to  Federal  money. 


INTEREST.  203 

Miscellaneous  Examples. 

25.  Reduce  187  £  8  s.  sterling  to  Federal  money. 

26.  Reduce  964  £  16  s.  sterling  to  Federal  money. 

27.  Reduce  1643  dollars  to  Sterling  money. 

28.  Reduce  1600  dollars  to  Sterling  money. 

29.  Reduce  167  £  14s.  Canada,  to  Federal  money. 

30.  Reduce  $196.50  to  Canada  currency. 

31.  Reduce  $1674.40  to  New  England  currency. 

32.  Reduce  $744.15  to  New  York  currency. 

33.  Reduce  142 £   14s.  6d.   New  York,    to    New 
England  currency. 

34.  Reduce  643  £  15  s.  9d.  Pennsylvania  currency 
to  Federal  money. 

35.  Reduce  $172.31  to  Pennsylvania  currency. 


SECTION    XXVIII. 

INTEREST. 
[See  Section  XIV.,  Part  I.] 

Rule.  —  Find  the  interest  of  1  dollar  for  the  given 
time ;  multiply  the  principal  by  it,  and  point  off  as  in 
the  multiplication  of  decimals. 

1.  What  is  the  interest  of  $156.34  for  11  months 

and  20  days  ? 


As  the  interest  of  1  dollar  for  2 
months  is  1  cent,  for  10  months  it 
will  be  5  cents,  .05.  As  the  inter- 
est of  1  dollar  for  6  days  is  1  mill, 
for  30  days  it  will  be  5  mills,  and 
for  20  days  3  mills  and  £,  making 
8  mills  and  £  Set  down  the  8  at  I  $9.11983,  Ans. 
the  right  hand  of  the  .05,  and  for  the  £  divide  by  3. 


3)156.34 

.058 

125072 

78170 
5211 


204  INTEREST. 

Observe,  the  0  before  the  5  must  be  retained ;  other- 
wise it  would  be  5  tenths  of  a  dollar,  or  50  cents,  and 
the  answer  would  be  10  times  too  great.  If  there  are 
no  cents,  there  must  be  two  ciphers  at  the  left  hand  of 
the  mills.  The  number  of  cents  for  the  multiplier  is 
always  equal  to  half  the  greatest  even  number  of 
months ;  the  number  of  mills  is  one  sixth  of  all  the 
days  over  and  above  the  greatest  even  number  of 
months. 

2.  Interest  of  $384.18  for  7  months  and  10  days? 

3.  Interest  of  $147.19  for  5  months  15  days? 

4.  Interest  of  $568.25  for  9  months  13  days? 

5.  Interest  of  $81.40  for  10  months  14  days? 

6.  Interest  of  $56.32  for  12  months  24  days? 

7.  Interest  of  $75.30  for  14  months  18  days? 

8.  Interest  of  $644.46  for  15  months  24  days? 

9.  Interest  of  $831.00  for  1   year,  4  months,  12 
days? 

10.  Interest  of  $380.00  for  1  year  7  months? 

11.  Interest  of  $500.00  for   1  year,  5  months,  6 
days? 

12.  Interest  of  $27.42  for  4  months  17  days? 

13.  Interest  of  $13.18  for  6  months  23  days? 

14.  Interest  of  $1000.00  for  5  months  4  days? 

15.  Interest  of  $65.48  for  30  days,  or  1  month? 

16.  Interest  of  $94.00  for  30  days? 

17.  Interest  of  $840.60  for  18  days? 

18.  Interest  of  $632.00  for  18  days? 

19.  Interest  of  $349.40  for  12  days? 

20.  Interest  of  $267.62  for  12  days? 

21.  Interest  of  $384.92  for  15  days? 

22.  Interest  of  $811.19  for  20  days  ? 

23.  Interest  of  $673.94  for  5  months  11  days? 

24.  Interest  of  $460.00  for  8  months  18  days? 

25.  Interest  of  $460.00  for  8  months  18  days,  at  12 
per  cent.  ? 

26.  Interest  of  $460.00  for  8  months  18  days,  at  8 
per  cent.  ? 


INTEREST.  205 

27.  Interest  of  $460.00  for  8  months  18  days,  at  7 
per  cent.  ? 

28.  Interest  of  $460.00  for  8  months  18  days,  at  5 
per  cent.  ? 

29.  Interest  of  $1500.00  for  15  months,  at  4  per 
cent.  ? 

30.  Interest  of  $145.80  for  7  months  11  days,  at  8 
per  cent.  ? 

31.  Interest  of  $341.18  for  2  years,  9  months,  18 
days? 

As  the  interest  of  a  dollar  for  30  days  is  5  mills,  for 
|  of  30  days,  or  6  days,  it  is  1  mill.  As  1  mill  is  TsW) 
one  thousandth  of  a  dollar,  it  follows  that  the  interest 
of  1  dollar  for  1  day  is  one  sixth  of  a  thousandth,  or 
stjW  °f  a  dollar.  For  two  days,  therefore,  it  will 
be  sttW,  for  1 5  days  s^vi  of  a  dollar. 

A  convenient  rule,  therefore,  when  the  time  is 
short,  is  the  following  :  — 

Multiply  the  sum  by  the  number  of  days,  and  divide 
the  product  by  6000. 

This  is  often  the  shortest  method.  You  divide  by 
1000,  by  removing  the  decimal  point  three  places  to 
the  left.  It  only  remains,  then,  after  doing  this,  to 
multiply  by  the  number  of  days,  and  divide  by  6. 

32.  What  is  the  interest  of  $348.25  for  18  days? 
Dividing   by   1000,  you   have    $0.348£,  thirty-four 

cents  eight  mills  and  a  quarter.  Instead,  now,  of  mul- 
tiplying by  18  and  dividing  by  6,  you  may  multiply 
by  3,  for  18  is  3  times  6. 

3  times  0.348]  is  $1.044f,  Arts. 

33.  Interest  of  $725.80  for  24  days? 

34.  Interest  of  $341.18  for  36  days? 

35.  Interest  of  $67.45  for  54  days? 

36.  Interest  of  $641.18  for  42  days? 

37.  Interest  of  $84.16  for  15  days? 

18 


206  PARTIAL    PAYMENTS. 


» 


To  find  the  amount,  add  the  interest  to  the  principal  j 
or,  find  the  amount  of  $  1.00  for  the  given  time,  and 
multiply  the  principal  by  it. 

38.  What  is  the  amount  of  $560.50  for  8  months 
12  days  ? 

39.  Amount  of  $964.25  for  15  months  18  days? 

40.  Amount  of  $460.00  for  1  year  6  months? 

41.  Amount  of  $120.50  for  2  years  4  months? 

42.  Amount  of  $68.40  for  1  year,  6  months,  24 
days? 

43.  Amount  of  $500.00  for  2  years  3  months? 

44.  Amount  of  $730.50  for  6  months  12  days? 

45.  Amount  of  $840.25  for  4  months  18  days? 

46.  Amount  of  $40.50  for  8  months  12  days? 


SECTION    XXIX. 

PARTIAL   PAYMENTS. 

When  partial  payments  are  made  on  a  note,  the 
amount  due  on  the  final  payment  of  the  note  may  be 
found  by  the  following  rule  :  — 

Find  the  interest  on  the  note  up  to  the  time  of  the 
first  payment.  If  the  payment  exceeds  the  interest, 
deduct  it  from  the  amount,  regarding  the  remainder  as 
a  new  principal.  On  this,  calculate  the  interest  to  the 
time  of  the  next  payment ;  and  so  on.  If  any  payment 
is  less  than  the  interest  then  due,  reserve  it,  and  com- 
pute the  interest  on  to  the  time  when  the  payments, 
added  together,  shall  exceed  the  interest  due.  Then 
subtract  the.  sum  of  the  payments  from  the  amount 
then  due,  and  proceed  as  before. 


PARTIAL    PAYMENTS.  207 

1.  A  note  of  200  dollars  is  given  July  1,  1834,  on 
which  are  the  following  partial  payments  :  — 

Dec.  15,  1834,  .  .  $25.00. 

March  1,  1835, 2.50. 

Aug.  10,  1835, .  .  .  45.00. 
What  was  due  Dec.  31,  1835  ? 

2.  A  note  of  $340.25  is  given  Aug.  1,  1840. 
Endorsements,  — Jan.  10,  1841,  .  .$28.40. 

July  1,  1841, 9.00. 

March  14,  1842, .  .  74.00. 
What  was  due  Jan.  1,  1843? 

3.  A  note  of  $480.00  is  given  June  9,  1841. 
Endorsements,  —  Sept.  11,  1842,  .  $60.00. 

Jan.  3,  1843, 95.00. 

March  12,  1844, .  100.00. 
What  was  due  Dec.  1,  1844  ? 

4.  A  note  of  $675.40  is  given  July  3,  1843. 
Endorsements,  — Jan.  4,  1844,.  .  .  $65.00. 

April  17,  1844,  .  .  29.50. 
Nov.  18,  1844,..  .  74.00. 
What  is  due  Jan.  1,  1845  ? 

5.  A  note  of  $345.40  is  given  April  1,  1843. 
Endorsements,  —  Dec.  1,  1843,.  .  .$40.00. 

-    June  10,  1844,  .  .  .  90.00. 

Oct.  4,  1844, 31.50. 

Feb.  6,  1845, 17.00. 

What  is  due  Jan.  1,  1846  ? 

The  rule  given  above  is  the  legal  rule.  When, 
however,  the  note  is  paid  within  a  year  from  the  time 
when'  it  was  given,  the  following  rule  is  usually  em- 
ployed :  — 

Find  the  amount  of  principal  and  interest  of  the 
whole  note,  from  the  time  it  was  siven  till   the  final 


208 


ANNUAL    INTEREST. 


payment ;  find  the  amount  of  each  payment,  from  the 
time  it  was  paid  till  the  final  payment ;  and  the  sum 
of  these  amounts  subtract  from  the  amount  of  the 
whole  note.     The  remainder  will  be  the  balance  due. 

6.  A  note  of  $525.00  is  given  Sept.  1,  1844. 
Endorsements,  — Dec.  30,  1844,  .  .  $58.75. 

March  4,  1845, .  .  104.20. 

June  8,  1845, ....  63.40. 
What  is  due  Aug.  21,  1845? 

'  7.  A  note  of  $784.50,  given  July  7,  1844,  has  the 
following  endorsements :  — 

Sept.  5,  1844,.  .  .$54.00. 

Nov.  10,  1844,  .  .  .  60.00. 

Jan.  12,  1845, 75.00. 

March  17,  1845,  .  100.00. 
What  is  due  May  1,  1845  ? 


ANNUAL   INTEREST. 

When  a  note  is  given  payable  at  a  longer  period 
than  a  year  from  the  date,  it  is  usual  to  express  in  the 
note  that  the  interest  shall  be  paid  annually.  At  the 
end  of  a  year,  the  holder  cf  the  note  may  compel  the 
payment  of  the  interest.  In  such  cases,  the  debtor, 
instead  of  paying  the  interest  that  is  due,  sometimes 
renews  the  note,  adding  the  interest  to  the  principal. 
Thus,  at  the  end  of  each  year,  the  interest  due  is  added 
in,  and  goes  to  make  a  new  principal  for  the  following 
year.  This  is  called  Compound  Interest;  but  the 
computation  of  it  is  the  same  as  in  simple  interest ; 
for,  if  the  interest  is  not  computed  every  year,  and 
either  paid  or  put  into  the  note  by  renewal,  that  in- 
terest cannot  draw  interest.*     The  law  regards  it  as 

*  In  some  of  the  states,  the  interest,  after  it  falls  due,  draws  simple 
interest  till  it  is  paid. 


ANNUAL    INTEREST.  209 

the  duty  of  the  creditor  to  remind  the  debtor  of  his 
debt,  by  exacting  the  payment  of  the  interest  every 
year.  If  he  does  not  do  this,  he  can  derive  no  advan- 
tage from  the  promise  in  the  note  to  pay  the  interest 
.  annually. 

ILLUSTRATION. 


$100.00.  Boston,  March  1,  1845. 

For  value  received,  I  promise  to  pay  to  John  Jones, 
or  order,  one  hundred  dollars,  in  five  years,  with  inter- 
est annually.  a  0 

J  Samuel  Barton. 

If  John  Jones  does  not  exact  the  interest  till  the 
end  of  the  five  years,  and  if  he  obtains  no  renewal  of 
it,  the  amount  of  the  note  will  be  only  $  130.00  ;  for 
the  interest  of  100  dollars,  for  five  years,  is  30  dollars. 

If,  however,  he  obtains  a  renewal  of  the  note  at  the 
end  of  each  year,  the  principal  of  the  note,  for  the 
second  year,  will  be  $106.00. 

8.  What  will  the  principal  of  the  note  for  the  third 
year  be  ? 

9.  What  will  the  principal  of  the  note  for  the  fourth 
year  be  ? 

10.  What  will  the  principal  of  the  note  for  the  fifth 
year  be  ? 

11.  What  will  be  due,  principal  and  interest,  at  the 
end  of  the  fifth  year? 

12.  How  much  would  the  holder  of  the  above  note 
lose  by  omitting  to  obtain  any  renewal  of  it,  or  any 
payment  of  annual  interest  ? 


$250.00.  New  York,  July  1,  1845. 

For  value  received,  I  promise  to  pay  John  Foss,  or 

order,  two   hundred  and  fifty  dollars,  in  four  years, 

with  interest  annually.  ,  ~ 

18*  Amos  Carr. 


210  DISCOUNT. 

13.  If  no  interest  is  paid  on  this  note  till  the  prin- 
cipal is  due,  and  if  no  renewal  of  the  note  is  made, 
what  will  be  the  amount  of  the  note  at  the  time  of 
payment  ? 

14.  If  the  note  is  renewed  each  year,  what  will  be 
the  principal  of  the  note  for  the  second  year  ? 

15.  What  will  be  the  principal  of  the  note  for  the 
third  year  ? 

16.  What  will  be  the  principal  of  the  note  for  the 
fourth  year  ? 

17.  What  will  be  the  amount  of  the  last  note  at 
the  time  of  payment  ? 

18.  How  much  would  the  holder,  John  Foss,  lose, 
by  neglecting  to  obtain  any  annual  payment  of  inter- 
est, or  any  renewal  of  the  above  note  ? 


SECTION    XXX. 

DISCOUNT. 

[See  Section  XIV.,  Part  I.] 

Examples. 

1.  What  is  the  present  worth  of  $475.50,  payable 
in  3  months  ? 

Rule.  —  Deduct  from  the  given  sum  its  interest  for 
the  specified  time.  The  remainder  will  be  the  present 
worth. 

2.  What  is  the  present  worth  of  $341.00,  payable 
in  65  days  ? 

3.  Present  worth  of  $940.25,  payable  in  4  months? 

4.  Present  worth  of  $156.30,  payable  in  96  days? 

5.  Present  worth  of  $312.60,  payable  in  35  days? 


BANKING. 


211 


6.  Present  worth  of  $500.00,  payable  in  41  days? 

7.  Present  worth  of  $814.67,  payable  in  65  days? 

8.  Present  worth  of  $46.30,    payable  in  20  days? 

9.  Present  worth  of  $124.45,  payable  in  5  months? 
10.  Present  worth  of  $360.20,  payable  in  4^  months  ? 


SECTION    XXXI. 

BANKING. 

[See  Section  XIV.,  Part  I.] 

To  find  the  present  worth  of  a  note  given  to  a 
bank,  payable  at  some  future  time,  find  the  present 
worth  of  1  dollar  for  the  given  time,  and  multiply 
the  sunt  named  in  the  note  by  it. 

1.  What  is  the  present  worth  of  a  note  for  100  dol- 
lars, discounted  at  a  bank,  for  60  days  ? 

Interest  of  1  dollar  for  63  days  is  .0105.  This 
subtracted  from  1  dollar,  leaves  for  the  present  worth 
.9895. 

2.  What  is  the  present  worth  of  a  note  for  $450.00, 
discounted  at  a  bank,  for  90  days  ? 

3.  I  give  my  note  to  a  bank  for  $250.00,  for  60 
days.     What  do  I  receive  ? 

4.  I  give  my  note  to  a  bank  for  $520.00,  for  120 
days.     What  do  I  receive  ? 

5.  Present  worth  of  a  bank  note  for  $600.00,  dis- 
counted for  60  days  ? 

6.  Present  worth  of  a  bank  note  for  $150.00,  dis- 
counted for  120  days  ? 

7.  Present  worth  of  a  bank  note  for  $75.00,  dis- 
counted for  30  days? 


212         LOSS  AND  GAIN. PER  CENTAGE. 

8.  Present  worth  of  a  bank  note  for -$1000.00, 
discounted  for  60  days? 

9.  Present  worth  of  a  bank  note  for  $560.00, 
discounted  for  120  days? 

10.  Present  worth  of  a  bank  note  for  $150.00, 
discounted  for  30  days? 

To  find  what  must  be  the  face  of  a  note  given  to 
a  bank,  in  order  to  obtain  a  certain  sum, — Find  the 
present  ivorth  of  1  dollar  for  the  given  time,  and  di- 
vide the  sum  you  wish  to  obtain  by  it.  The  quotient 
will  express  the  sum  that  must  be  named  in' the  note. 
This,  you  observe,  is  just  the  reverse  of  the  preceding 
case. 

11.  For  what  sum  must  I  give  my  note  to  a  bank, 
payable  in  60  days,  in  order  to  receive  $98.95? 

12.  For  what  sum  must  I  give  my  note  to  a  bank, 
payable  in  120  days,  in  order  to  receive  $509.34? 

13.  For  what  sum  must  I  give  my  note  to  a  bank, 
payable  in  60  days,  in  order  to  receive  $593.70? 

14.  For  what  sum  must  I  give  my  note  to  a  bank, 
payable  in  30  days,  in  order  to  receive  $10000.00? 


SECTION    XXXII. 

LOSS   AND    GAIN.  — PER   CENTAGE. 

[See  Section  XIV.,  Part  I.] 

1.  A  man  bought  a  horse  for  75  dollars,  and  sold 
him  for   $82.50.     What  did  he  gain  per  cent.? 

2.  A  man  bought  a  chaise  for  $178.00,  and  sold  it 
for  $154.50.     What  did  he  lose  per  cent.  ? 

3.  A  merchant  bought  a  lot  of  flour  at   $4.62  a 


PER    CENTAGE.  213 

barrel,  and    sold   it    at    $5.15   a  barrel.     What    was 
his  gain  per  cent.  ? 

4.  A  merchant  bought  a  piece  of  broadcloth  for 
$4.30  per  yard.  What  must  he  sell  it  for  to  gain 
12  per  cent.  ? 

5.  A  man  has  $1200.00  invested  in  a  manufactory. 
He  receives,  for  his  half-yearly  dividend,  30  dollars. 
What  per  cent,  is  that  on  his  stock  ? 

6.  A  merchant  fails,  owing  $8540.00,  and  can  pay 
but  $2700.00.     How  much  will  that  be  on  a  dollar? 

7.  A  man,  failing  in  business,  agrees  to  pay  his 
creditors  87  cents  on  a  dollar.  What  must  a  cred- 
itor receive  whose  claim  is   $740.30? 

The  pupil  should  be  encouraged  habitually  to  reason 
upon  the  operations  he  performs ;  so  that  his  method 
of  procedure  may  be  suggested  by  the  relations  of  the 
numbers,  and  not  dictated  by  a  special  rule.  To  aid 
in  this  important  habit,  a  few  remarks  will  be  made 
on  some  of  the  foregoing  examples.  These  may 
serve  as  specimens  of  analysis,  and  suggest  to  the 
student  a  similar  course  of  reasoning  in  other  cases. 

Example  1.  The  whole  gain  is  $7.50.  If  this 
gain  were  made  on  an  outlay  of  1  dollar,  the  gain 
would  be  seven  hundred  and  fifty  per  cent.  But  the 
gain  is  made  on  an  outlay  of  75  dollars.  The  gain  per 
cent.,  therefore,  is  one  seventy-fifth  of  the  whole  gain. 

Example  4.  If  the  cost  was  1  dollar  a  yard,  he 
must  add  12  cents ;  if  2  dollars,  he  must  add  24 
cents;  &c. 

Example  5.  If  30  dollars  had  been  the  gain  upon 
1  dollar,  it  would  have  been  30  hundred  per  cent. 
But  the  gain  was  upon  1200  dollars.  The  per  cent., 
therefore,  must  be  one  twelve-hundredth  of  30  dollars. 

8.  I  invest  in  a  factory  1260  dollars,  and  receive 
for  my  yearly  dividend  86  dollars.  What  is  that 
per  cent.? 


214  PER    CENTAGC. 

9.  I  purchase  flour  at  $4.75  per  barrel.  What 
must  I  sell  it  for  to  gain  12  per  cent.? 

10.  A  merchant  bought  a  ship  for  11475  dollars,  and 
sold  her  for  $13680.     What  did  he  gain  per  cent.  ? 

11.  The  population  of  the  state  of  New  York,  in 
1810,  was  959949.  In  1820,  it  was  1372812.  What 
was  the  gain  per  cent,  in  that  term  of  10  years? 

12.  In  1830,  it  was  1918604.  What  was  the  gain 
per  cent,  from  1820  to  1830? 

13.  In  1840,  it  was  2428921.  What  was  the  gain 
per  cent,  from  1830  to  1840? 

14.  The  population  of  Ohio,  in  1810,  was  230760. 
In  1820,  it  was  581434.  What  was  the  gain  per 
cent.  ? 

15.  The  population  of  Ohio,  in  1830,  was  937903. 
What  was  the  gain  per  cent,  from  1820  to  1830? 

16.  In  1840,  it  was  1519467.  What  was  the  gain 
per  cent,  from  1830  to  1840? 

17.  Massachusetts  had,  in  1810,  472040  inhabitants. 
In  1820,  it  had  523287.  What  was  the  gain  per  cent, 
in  10  years? 

18.  Massachusetts  had,  in  1830,  610408  inhabitants. 
What  was  her  gain  per  cent,  from  1820  to  1830? 

19.  In  1840,  Massachusetts  had  737699  inhabitants. 
What  was  the  gain  per  cent,  from  1830  to  1840? 

20.  An  agent  sells  12000  dollars'  worth  of  cloth  for 
a  factory,  charging  2|  per  cent,  commission.  What 
will  be  his  remuneration? 

21.  If  I  buy  for  a  merchant,  at  a  commission  of  4 
per  cent.,  500  barrels  of  flour,  at  $4.40  per  barrel, 
what  am  I  entitled  to  for  my  commission  ? 

22.  What  is  3  per  cent,  on  $674.54? 

23.  What  is  2  per  cent,  on  $781.50? 

24.  What  is  the  value  of  five  100  dollar  shares  in  a 
bank,  at  4£  per  cent,  advance? 

25.  What  is  the  value  of  seven  100  dollar  shares, 
at  6  per  cent,  discount  ? 


PER    CENTAGE.  215 

26.  What  is  the  value  of  18  shares  bank  stock,  60 
dollars  a  share,  at  4  per  cent,  discount  ? 

27.  What  is  the  duty  on  a  quantity  of  broadcloth, 
whose  value  is  1735  dollars,  at  15  per  cent.  ? 

28.  What  is  the  duty  on  a  quantity  of  iron,  whose 
value  is  3456  dollars,  at  18  per  cent.? 

29.  What  is  the  commission  on  the  sale  of  1246 
dollars'  worth  of  cloth,  at  3  per  cent.  ? 

30.  A  man  bought  a  lot  of  hay  for  13  dollars  a  ton. 
He  sold  it  for  $  14.25  a  ton.  What  did  he  gain  per 
cent.  ? 

31.  Bought  tea  for  46  cents  a  pound.  What  must 
I  sell  it  for  a  pound  to  gain  12  per  cent.  ? 

32.  What  is  the  worth  of  750  dollars,  bank  stock, 
at  7£  per  cent,  advance  ? 

33.  What  is  the  worth  of  8500  dollars,  bank  stock, 
at  9  per  cent,  discount? 

34.  I  sell  flour  at  $5.32  per  barrel,  and  thereby 
gain  12  per  cent,  on  my  outlay.  What  did  the  flour 
cost? 

Every  $1.00  laid  out  in  the  purchase  has  brought 
me  a  return  of  $1.12.  The  number  of  dollars  I  paid 
out  on  a  barrel  must  therefore  equal  the  number  of 
times  $1.12  will  go  in  $5.32. 

35.  A  merchant  sells  a  ship  for  13680  dollars,  gain- 
ing thereby  14-&  per  cent,  on  what  she  cost  him. 
What  did  the  ship  cost  ? 

36.  300  dollars  is  2£  per  cent,  on  what  sum? 

37.  $15.63  is  2  per  cent,  on  what  sum? 

38.  Bought  12  barrels  of  flour,  each  containing  196 
pounds,  at  $5.42  per  barrel,  and  sold  it  at  26  cents  for 
7  pounds.  How  much  gain  in  the  whole,  and  how 
much  gain  per  cent.? 

39.  Bought  43  dozen  pairs  of  shoes,  at  $4.30  per 
dozen,  and  sold  them  at  62  cents  per  pair.  What  gain 
in  all?    What  gain  per  cent.? 


216  ALLIGATION. 

40.  Bought  20  barrels  of  apples,  each  containing 
2f  bushels,  at  $2.10  per  barrel,  and  sold  them  at 
$1.25  per  bushel.  What  gain  in  all?  What  gain 
per  cent.? 

41.  Bought  375  barrels  of  flour,  at  $5.20  per 
barrel,  and  sold  200  barrels  at  $6.10;  the  remainder 
at  $6.42  per  barrel.  What  gain  in  all?  What  gain 
per  cent.  ? 

42.  Bought  34  acres  of  land,  at  41  dollars  per  acre. 
Sold  it  for  $1700.00.  How  much  gain  in  all?  What 
gain  per  cent.  ? 


SECTION    XXXIII. 

ALLIGATION.* 

The  operations  under  this  rule  show  the  method  of 
rinding  the  value  of  a  mixture,  when  the  price  and 
quantity  of  each  of  its  ingredients  are  given ;  also,  to 
find  the  quantity  of  each  ingredient,  when  its  price  is 
given,  and  it  is  required  to  unite  them  so  as  to  form  a 
mixture  of  a  given  value. 

Case  1. —  To  find  the  value  of  the  mixture,  when 
the  quantity  and  -price  of  each  of  the  ingredients 
are  given. 

1.  Mix  15  bushels  of  oats,  at  40  cents  per  bushel ; 
12  bushels  of  barley,  at  60  cents;  and  24  bushels  of 


*  The  word  alligation  signifies  a  tying  together,  and  has  reference 
to  a  particular  way  of  linking  numbers  together,  by  means  of  which 
operations  of  this  kind  have  been  performed.  The  name  is  retained 
as  a  matter  of  convenience ;  but  I  have  thought  it  best  for  the  prog- 
ress of  the  pupil,  that  he  should  pursue  a  strictly  analytical  method 
in  all  the  operations. 


ALLIGATION.  217 

corn,  at  83  cents.     What  will  the  mixture  be  worth 
per  bushel? 

It  is  evident  that,  if  you  find  the  value  of  the  whole, 
and  divide  the  sum  by  the  number  of  bushels,  the 
quotient  will  be  the  value  per  bushel. 

r  2.  Mix  20  pounds  of  tea,  at  43  cents  per  pound  ; 
18  lbs.  at  61  cents ;  and  11  lbs.  at  74  cents  per  pound. 
What  will  the  mixture  be  worth? 

3.  If  41  lbs.  of  coffee,  at  13  cents  per  lb.,  be  mixed 
with  45  lbs.  at  9£  cents,  and  27  lbs.  at  15  cents, 
what  will  the  mixture  be  worth  per  pound? 

Case  2.  —  To  find  the  quantity  of  each  ingredient, 
when  its  price  and  that  of  the  required  mixture  are 
given. 

4.  If  I  mix  oats,  worth  2  s.  per  bushel,  with  rye, 
worth  5  s.,  so  as  to  make  the  mixture  worth  3  s.  per 
bushel,  in  what  proportion  must  I  mix  them  ? 

It  is  evident,  that,  if  I  put  in  1  bushel  of  oats,  I 
gain  1  shilling.  Now,  I  must  put  in  rye  enough  with 
this  bushel  of  oats  to  lose  1  shilling.  On  every  bushel 
of  rye  put  in,  I  lose  2  shillings ;  therefore,  in  order  to 
lose  1  shilling,  I  must  put  in  £  a  bushel.  I  must 
therefore  put  in  1  bushel  of  oats  to  £  a  bushel  of  rye. 
It  is  evident  that,  if  I  double  the  quantity  thus  found 
of  each  ingredient,  the  value  of  the  mixture  will  be 
the  same ;  or  I  may  take  any  equal  multiples  of  the 
quantities,  as  4  bushels  of  oats  and  2  bushels  of  rye, 
6  bushels  of  oats  and  3  bushels  of  rye,  20  bushels  of 
oats  and  10  bushels  of  rye,  &c. 

5.  If  I  mix  oats,  worth  2  s.  per  bushel,  with  rye, 
worth  6  s.,  so  as  to  make  the  mixture  worth  3  s.  per 
bushel,  in  what  proportion  must  they  be  mixed  ? 

6.  Mix  oats,  worth  3  s.  per  bushel,  with  wheat, 
worth  7  s.,  so  as  to  make  the  mixture  worth  5  s.  per 
bushel.     In  what  proportion  must  they  be  mixed  ? 

19 


218  ALLIGATION. 

7.  Mix  the  same  ingredients,  at  the  same  price,  so 
as  to  make  the  mixture  worth  6  s.  per  bushel.  In 
what  proportion  must  they  be  mixed? 

8.  In  what  proportion  must  oats,  worth  2  s.,  and 
wheat,  worth  8  s.,  be  mixed,  to  make  the  mixture 
worth  4s.  per  bushel? 

9.  How  can  you  mix  corn,  worth  80  cents  per  bushel, 
and  rye,  worth  85  cents,  with  barley,  worth  46  cents, 
so  as  to  make  a  mixture  worth  60  cents  per  bushel  ? 

Here  you  have  three  ingredients.  First,  mix  barley 
with  one  of  the  dearer  ingredients,  so  as  to  make  a 
mixture  of  the  required  value.  Then  mix  barley 
with  the  other  ingredient,  and  see  how  much  you 
have  taken  of  each. 

10.  Mix  3  sorts  of  tea,  at  25  cents,  33  cents,  and  40 
cents,  per  pound,  so  as  to  make  a  mixture  worth  30 
cents  per  pound. 

11.  Mix  tea  at  20  cents,  with  tea  at  45  cents,  and 
tea  at  54  cents,  per  pound,  so  as  to  make  a  mixture 
worth  38  cents  per  pound. 

12.  If  you  mix  sugar,  at  6  cents,  8  cents,  10  cents, 
and  11  cents,  per  pound,  in  what  quantities  may  they 
be  taken  so  as  to  make  a  mixture  worth  9  cents  per 
pound  ? 

First,  take  two  of  the  ingredients,  one  cheaper  and 
one  dearer  than  the  mixture.  Form  a  mixture  of 
these.  Then  take  the  two  remaining  ingredients  in 
the  same  way. 

13.  If  three  sorts  of  spirit,  worth  60  cents,  75  cents, 
and  80  cents,  per  gallon,  are  mixed  with  water,  costing 
nothing,  what  must  be  the  proportion  to  make  a  mix- 
ture worth  70  cents  per  gallon  ? 

It  is  immaterial  in  what  way  you  select  the  pairs  of 
ingredients,  provided,  in  each  pair,  one  of  the  ingre- 
dients   be   cheaper  and   the   other   dearer   than   the 


ALLIGATION.  219 

required  mixture.  Thus  a  great  variety  of  answers 
may  be  obtained  whenever  there  is  more  than  one 
pair  of  ingredients.  In  all  cases,  however,  the  cor- 
rectness of  the  operation  may  be  proved  in  the  follow- 
ing way :  — 

Find  the  total  value  of  all  the  ingredients.  If  this 
is  equal  to  the  value  of  the  whole  mixture  at  the 
required  price,  the  work  is  right. 

14.  Mix  5  sorts  of  grain,  at  25  cents,  30  cents,  33 
cents,  45  cents,  and  50  cents,  so  as  to  make  a  mixture 
worth  40  cents,  per  bushel. 

Case  3.  —  When  the  quantity  of  one  ingredient  is 
given. 

15.  Mix  brandy,  at  74  cents  per  gallon,  with  24 
gallons  of  brandy,  at  1  dollar  per  gallon,  so  that  the 
mixture  may  be  worth  80  cents  per  gallon. 

Here  you  observe  that  the  quantity  of  one  of  the 
ingredients  is  given.  We  will  first  make  a  mixture  of 
the  two,  without  regard  to  this  circumstance.  If  I  put 
in  1  gallon  at  1  dollar,  I  lose  20  cents.  For  every 
gallon  at  74  cents,  which  is  put  in,  I  gain  6  cents.  In 
order  to  gain  20  cents,  I  must,  therefore,  put  in  3-£ 
gallons.  The  quantities  stand,  then,  1  gallon  at  1 
dollar,  3£  gallons  at  74  cents.  But  I  wish  to  put  in 
24  gallons  at  1  dollar.  To  balance  this,  I  must  there- 
fore put  in  24  times  3£  gallons  at  74  cents  ;  that  is,  80 
gallons. 

16.  Mix  sugar,  at  8  cents,  11  cents,  and  12  cents, 
with  100  lbs.  of  sugar  at  7  cents,  so  as  to  make  the 
mixture  worth  10  cents,  per  pound. 

Case  £.  —  When  the  quantity  of  the  required  mix- 
ture is  given. 

17.  Mix  oats,  at  40  cents,  with  corn,  at  60  cents,  so 


220  EQUATION    OF    PAYMENTS. 

as  to  form  a  mixture  of  100  bushels,  worth  48  cents 
per  bushel. 

If  I  put  in  1  bushel  at  40  cents,  I  gain  8  cents ;  if  I 
put  in  1  bushel  at  60  cents,  I  lose  12  cents.  To  lose 
8  cents,  therefore,  I  must  put  in  only  f  of  a  bushel. 
The  quantities  are,  then,  1  bushel  at  40  cents,  §  of  a 
bushel  at  60  cents ;  making,  when  added,  If  bushels. 
But  100  bushels  is  the  quantity  required.  100-r-$  = 
60.  Each  ingredient,  therefore,  must  be  multiplied  by 
60.  60X1  =  60;  60Xf  =  40.  The  quantities,  then, 
are  60  bushels  at  40  cents,  and  40  bushels  at  60  cents. 


SECTION    XXXIV. 


EQUATION   OF   PAYMENTS. 

If  A  owes  B  several  sums  of  money,  to  be  paid  at 
different  times,  he  may  desire  to  pay  the  whole  at 
once,  and  consequently  to  know  at  what  time  the 
whole  becomes  due.  This  time  is  found  by  making 
an  equation  of  the  payments,  multiplied  by  the  time, 
as  follows :  — 

1.  A  owes  B  200  dollars;  100  due  Jan.  1,  1844; 
100  due  Jan.  1,  1846.  He  wishes  to  pay  it  all  at 
once.     At  what  time  should  he  pay  it  ? 

Now,  on  Jan.  1,  1844,  A  is  entitled  to  the  use  of 
100  dollars  for  2  years  longer;  100X2  =  200;  equal 
to  the  use  of  1  dollar  for  200  years.  If  he  is  to  pay 
the  whole  together,  he  must  keep  the  200  dollars  long 
enough  to  balance  that  claim.  200  )  200  ( I  year,  the 
answer.  The  whole  should  be  paid  one  year  from 
Jan.  1,  1844. 


EQUATION    OF    PAYMENT.  221 

2.  A  owes  B  100  dollars  due  hi  6  monLhi,,  200  dol- 
lars due  in  12  months.  In  how  many  montns  should 
the  whole  be  paid  together  ? 

100  X  6=   600 
200X12^2400 


300:  300)3000 

10  months,  —  the  answer. 

The  above  is  the  method  usually  employed,  and  is 
sufficiently  exact  for  the  necessities  of  business  ;  but  it 
gives  a  result  a  little  in  favor  of  the  debtor ;  that  is,  it 
makes  the  equated  time  a  little  later  than  it  should  be. 
To  find  the  exact  equated  time,  is  a  problem  far  too 
difficult  to  be  used  in  ordinary  business. 

Rule. — Multiply  each  payment  by  the  length  of 
time  before  it  becomes  due.  Divide  the  sum  of  the 
products  by  the  sum  of  all  the  payments.  The  quo- 
tient will  express  the  length  of  time  in  which  the 
whole  is  due. 

3.  A  owes  B  several  sums,  due  at  different  times,  as 
follows:  $600  in  2  months,  $150  in  3  months,  $75 
in  6  months.  What  is  the  equated  time  for  the 
whole  ? 

4.  A  man  owes  $1000;  of  which  200  are  to  be 
paid  in  3  months,  400  in  6  months,  and  the  remainder 
in  8  months.  What  is  the  equated  time  for  the  pay- 
ment of  the  whole  ? 

5.  If  I  owe  $1200,  one  half  to  be  paid  in  3 
months,  one  third  in  6  months,  and  the  remainder  in 
9  months,  in  what  time  should  the  whole  be  paid  ? 

6.  A  owes  B  $640;  150  due  in  30  days,  200  due 
in  60  days,  and  the  remainder  in  90  days.  What  is 
the  equated  time  for  the  whole  ? 

7.  A  merchant  buys  goods  to  the  amount  of  $1800; 
one  third  to  be  paid  in  30  days,  one  third  in  45  days, 

19* 


222  SQUARE    MEASURE. 

and  the  remainder  in  90  days.     What  is  the  equated 
time  for  the  whole  ? 

8.  If  I  owe  $1000,  half  to  be  paid  in  60  days,  and 
half  in  120  days,  and  if  I  pay  $  100  down,  what  will 
be  the  equated  time  for  the  remainder? 


SECTION    XXXV. 

SQUARE  MEASURE. 

[See  Section  XV.,  Part  L] 

1.  There  is  a  field  in  the  form  of  a  square,  15  rods 
on  a  side.     How  many  square  rods  does  it  contain  ? 

2.  If  the  square  be  15£  rods  on  a  side,  how  many 
square  rods  will  it  contain  ? 

3.  How  many  square  rods  are  there  in  a  square  field 
measuring  17  rods  on  a  side  ? 

4.  If  the  field  measure  17£  rods  on  a  side,  how 
many  square  rods  will  it  contain  ? 

5.  What  is  the  contents  of  a  square  field  measuring 
2l£  rods  on  a  side  ? 

6.  What  is  the  area  of  a  rectangular  field,  its  length 
being  64  rods,  and  its  breadth  12£  rods  ? 

7.  There  is  a  rectangular  field,  its  dimensions  being 
24^  rods  and  76£  rods.     What  is  the  area  ? 

8.  How  many  acres  are  there  in  a  rectangular  field, 
its  dimensions  being  94  rods  and  76£  rods  ? 

9.  There  is  a  rectangular  field  containing  7  acres. 
Its  length  is  35  rods.     What  is  its  breadth  ? 

10.  There  is  a  rectangular  farm,  its  length  being 
132  rods,  its  breadth  86£.  How  many  acres  does  it 
contain  ? 

11.  There  is  a  rectangular  lot  of  land  containing 
325  acres.  It  measures  on  one  side  176  rods.  What 
will  it  measure  on  the  other  ? 


SQUARE    MEASURE. 


223 


12.  There  is  a  board  containing  12  square  feet.  It 
is  13  inches  wide.     How  long  is  it  ? 

13.  A  table  contains  15  square  feet.  It  is  4  feet 
long.     How  wide  is  it  ?  • 

14.  A  certain  room  contains  30  square  yards.     It  is 
,16  feet  wide.     How  long  is  it? 

*      15.  A  piece  of  cloth  is  If  yards  wide.     How  much 
in  length  will  it  require  to  make  8  square  yards? 

16.  There  is  a  room  15  feet  by  18.  How  many 
yards  of  carpeting,  £  of  a  yard  wide,  will  it  require  to 
cover  it? 

17.  How  many  feet  of  boards  will  it  require  to 
cover  the  sides  and  ends  of  a  barn,  as  high  as  to  the 
eaves,  (its  length  is  42  feet,  width  34,  and  height  18,) 
allowing  one  fifth  of  the  boards  to  be  wasted  in 
cutting  ? 

18.  What  will  the  above-named  amount  of  boards 
cost  at  $11.50  a  thousand  feet? 

19.  A  road,  3£  rods  wide,  passes  through  a  man's 
land  1  mile.     How  much  of  his  land  does  it  take? 

20.  To  what  damages  will  he  be  entitled,  allowing 
him  28  dollars  an  acre  ? 

21.  There  is  a  right-angled  triangle.  Its  base  is  64 
rods,  and  perpendicular  20  rods.  How  many  acres 
does  it  contain?     (See  Sec.  XVII.,  Pt.  I.) 

22.  There  is  a  right-angled  triangle.  Its  base  is  84 
rods,  and  perpendicular  26  rods.  How  many  acres 
does  it  contain? 

23.  There  is  a  right-angled  triangle.  Its  base  is  49 
r  rods,  perpendicular  34  rods.     How  many  acres  does  it 

contain  ? 

24.  There  is  a  right-angled  triangle.  Its  area  is 
640  rods;  the  base  is  64  rods.  What  is  the  per- 
pendicular ? 

25.  A  right-angled  triangle  has  an  area  of  1092 
rods.  Its  base  is  60  rods.  What  is  the  perpendic- 
ular ? 


224 


DUODECIMALS. 


SECTION    XXXVI. 


DUODECIMALS. 


In  measuring  wood  and  lumber,  the  dimensions  are 
taken  in  feet  and  inches.  As  one  inch  is  TV  of  a  foot, 
the  multiplication  of  feet  and  inches  by  feet  and 
inches  is  the  same  as  multiplying  integers  and  twelfths 
by  integers  and  twelfths.  Take  the  following  ex- 
ample :  — 


1.  What  is  the  contents  of  a 
board  3  feet  7  inches  long,  and 
2  feet  4  inches  wide  ? 


Operation. 

A 


A 


6  if 


28 
TI1> 


.  Ans.   6      ff     T2¥V 


This  answer  may  be  reduced  to  more  simple  terms. 
t^  —  ^  +  i^t]  adding  T22-  +  ff  =  ff,  and  this  again 
3=  2  feet  +  TV ;  adding  the  2  feet  to  the  6  feet,  the 
answer  stands,  8  feet-f-y^-f-Tfr- 

As  the  fractions  decrease  in  value  at  a  twelve-fold 
rate,  whenever  the  numerator  exceeds  12,  the  excess 
may  be  set  down,  and  the  1  or  more  carried  to  the 
next  higher  fraction. 


2.  Multiply  5  feet  2  inches 
by  11  feet  9  inches. 


5 

A 

11 

A 

3 

« 

Tfj. 

56 

« 

tIws.   60     A     T|¥. 


To  render  the  operation  more  simple,  call  the  12ths 
or  inches,  primes,  (marked',)  and  the  144ths  or  frac- 
tions of  the  jsecond  order,  seco?ids,  (marked";)  then 
begin  with  the  lowest  order  and  multiply,  setting  each 


DUODECIMALS.  225 


product  in^its  own  place,  with  the  mark  appropriate 
to  express  its  value. 


3.  Multiply  13  ft.  5  in.  by  2 
ft.  11  in. 


ft. 
13     5 
2  11 


12     3 
26  10 


I    Ans.    39     1'    7" 

4.  Multiply  3  ft.  9  in.  by  7  ft.  4  in. 

5.  Multiply  9  ft.  8  m.  by  4  ft.  9  in. 

6.  Multiply  15  ft.  2  in.  by  9  ft.  1  in. 

7.  Multiply  8  ft.  6  in.  by  2  ft.  4  in. 

8.  What  is  the  contents  of  a  board  14  ft.  5  in.  long, 
and  1  ft.  1  in.  wide? 

9.  How  many  feet  in  a  load  of  wood  8  ft.  6  in. 
long,  4  ft.  2  in.  wide,  and  3  ft.  7  in.  high  ? 

Multiply  two  of  the  dimensions  together,  and  that 
product  by  the  third  dimension. 

10.  How  much  wood  in  a  load  11  ft.  3  in.  long,  4 
ft.  4  in.  wide,  3  ft.  11  in.  high? 

Divide  the  cubic  feet  by  128  for  cords,  and  the  re- 
mainder by  16  for  cord  feet,  or  eighths  of  a  cord. 

11.  How  much  wood  in  a  pile  38  ft.  6  in.  long,  4 
ft.  2  in.  wide,  and  4  ft.  high? 

12.  How  much  wood  in  a  load  9  ft.  4  in.  long,  4  ft. 
3  in.  wide,  3  ft.  8  in.  high? 

13.  How  much  wood  in  a  load  7  ft.  8  in.  long,  4  ft. 

2  in.  wide,  3  ft.  4  in.  high  ? 

14.  How  much  wood  in  a  load  8  ft.  2  in.  long,  4  ft. 
wide,  4  ft.  3  in.  high? 

15.  How  many  cords  of  wood  will  a  shed  contain, 
whose  dimensions  inside  are  22  ft.  6  in.  long,  10  ft.  6 
in.  wide,  7  ft.  8  in.  high? 

16.  Three  men  own  equal  shares  in  a  lot  of  wood 
lying  in  two  piles.     One  pile  is  13  ft.  4  in.  long,  4  ft. 

3  in.  wide,  4  ft.  4  in.  high.     The  other  pile  is  17  ft. 

F 


226 


EXTRACTION  OF  THE  SQUARE  ROOT. 


long,  4  ft.  wide,  3  ft.  10  in.  high.     How  much  wood 
is  each  man's  share? 
See  note  on  page  105. 


SECTION     XXXVII 


EXTRACTION  OF  THE  SQUARE  ROOT. 

[See  Section  XVI.,  Part  I.] 

This  operation  will  be  best  understood  by  taking 
first  the  simplest  case,  where  the  number  is  an  exact 
square,  and  the  root  containing  only  two  figures. 


What  is  the  square 
root  of  196? 


Operation. 

i% 

100 


20 


96 

80 
16 

96 

00 


10,  1st  part  of  the  root. 
4,  2d  part  of  the  root. 

14,  Ans. 


Place  a  period  over  the  unit  figure ;  another  over 
that  of  hundreds.  This  will  show  how  many  fig- 
ures there  will  be  in  the  root;  for  the  square  of  a 
number  has  always  either  twice  as  many  figures  as 
the  number,  or  one  less  than  twice  as  many.  Find 
the  greatest  square  of  tens  in  the  first  period,  (in  the 
given  example,  100,)  and  set  its  root  (10)  in  the 
quotient.  This  will  be  the  first  part  of  the  root. 
Square  the  root.  Subtract  the  square  from  the  first 
period,  and  bring  down  the  figures  of  the  next  period 
for  a  dividend.     To   the  left   hand,  place  double   the 


EXTRACTION  OF  THE  SQUARE  ROOT.       227 

part  of  the  root  already  found  for  a  trial  divisor.  Find 
by  trial  what  the  next  figure  of  the  root  must  he,  and 
set  it  down  under  the  first  part  of  the  root.  This  is 
the  second  part,  or  unit  figure  of  the  root.  [In  try- 
ing for  this  figure,  remember  that  it  must  be  so  small, 
that,  when  the  divisor  shall  be  multiplied  by  it,  and 
the  square  of  itself  shall  be  added  to  the  product,  the 
sum  shall  not  exceed  the  dividend.]  Multiply  the 
divisor  by  the  new  figure  of  the  root;  to  this  add  the 
square  of  the  same  figure,  and  subtract  the  sum  from 
the  dividend.  If  the  number  is  an  exact  square  of 
two  periods,  as  in  the  above  example,  there  will 
be  no  remainder;  and  the  two  parts  of  the  root  thus 
found,  when  added  together,  will  give  the  whole  root. 

2.  What  is  the  square  root  of  225?  Of  324? 

3.  What  is  the  square  root  of  289?  Of  529? 

4.  What  is  the  square  root  of  361?  Of  729? 

5.  What  is  the  square  root  of  625?  Of  1024? 

6.  What  is  the  square  root  of  784?  Of  1296? 

7.  What  is  the  square  root  of  841?  Of  1849? 

8.  What  is  the  square  root  of  961?  Of  2601?  * 

If  there  are  more  than  two  periods,  first  consider 
only  the  two  left-hand  periods,  and  find  their  root  as 
above  directed ;  then  consider  the  part  of  the  root  ex- 
pressed by  these  two  figures  as  the  first  part  with 
reference  to  the  next  figure,  (to  indicate  this,  you 
must  annex  a  cipher,)  and  work  for  the  next;  and 
so  on. 

9.  W^hat  is  the  square  root  of  15625  ? 

10.  What  is  the  square  root  of  60516  ?  Of  104976? 
Of  213444? 

Square  Root  of  a  Decimal. 

If  there  are  decimals  in  the  number,  point  off  each 
way  from  the  place  of  units;  adding  a  cipher,  if  ne- 
cessary, to  make  the  right-hand  period  complete. 


228       EXTRACTION  OF  THE  SQUARE  ROOT. 

11.  What  is  the  square  root  of  2.56  ?     Of  12.25? 

12.  What  is  the  square  root  of  2.25?     Of  20.25? 

13.  What  is  the  square  root  of  156.25?    Of  132.25  ? 

14.  What  is  the  square  root  of  13.69?     Of  21.16? 

15.  What  is  the  square  root  of  88.36  ?     Of  53.29  ? 

16.  What  is  the  square  root  of  1.69?     Of  1.44? 

17.  What  is  the  square  root  of  .81  ?     Of  .64? 

18.  What  is  the  square  root  of  .01  ?     Of  6.25? 

Square  Root  of  a    Vulgar  Fraction. 

To  obtain  the  square  root  of  a  vulgar  fraction,  find 
the  square  root  of  the  numerator,  and  of  the  denomi- 
nator, and  write  the -former  over  the  latter. 

19.  What  is  the  square  root  of  |  ?     Of  £f  ? 

20.  What  is  the  square  root  of  T\  ?     Of  A  ? 

21.  What  is  the  square  root  of  £ff  ?     Of  ff  ? 

22.  What  is  the  square  root  of  fff  ?     Of  fff  ? 
The  correctness  of  the  answer  may  always  be  tested 

by  multiplying  the  answer  found,  by  itself.     If  cor- 
rect, it  will  reproduce  the  original  square. 

23.  What  is  the  square  root  of  i?     Of  *V? 

24.  What  is  the  square  root  of  £f  ?     Of  tVt  ? 

Another  Method  of  finding  the  Root  of  a  Fraction. 
—  Reduce  the  fraction  to  a  decimal,  and  proceed  as 
already  directed  in  the  case  of  decimals. 

25.  What  is  the  square  root  of  i  ?  The  square  root 
of  1  is  1 ;  the  square  root  of  4  is  2.  Ans.  £.  Or  re- 
duce i  to  a  decimal,  — .25;  square  root,  .5,  Ans. 

If  the  number  is  not  a  complete  square,  annex 
periods  of  ciphers,  as  decimals,  and  carry  the  opera- 
tion as  far  as  desired. 

26.  What  is  the  square  root  of  70  ?     Of  80  ? 

27.  What  is  the  square  root  of  90  ?     Of  45  ? 

28.  What  is  the  square  root  of  60  ?     Of  84  ? 

29.  What  is  the  square  root  of  200?     Of  120? 


EXTRACTION  OF  THE  SQUARE  ROOT.       229 

30.  There  is  a  field  in  the  form  of  a  square,  contain- 
ing 1  acre.     How  many  rods  does  it  measure  on  a  side  ? 

31.  There  is  a  right-angled  triangle,  its  hypote- 
nuse measuring  60  rods.  What  is  the  sum  of  the 
squares  of  the  two  legs?     (See  Sec.  XVII.,  Part  I.) 

32.  There  is  a  right-angled  triangle.  The  squares 
of  its  legs  added  together  are  81  rods.  What  is  the 
length  of  the  hypotenuse  ? 

33.  There  is  a  right-angled  triangle.  Its  legs  meas- 
ure, one  25,  the  other  30  rods.  How  long  is  the  hy- 
potenuse ? 

34.  Two  men  start  from  the  same  place.  One 
travels  8  miles  east  j  the  other,  15  miles  north.  How 
far  are  they  then  apart  ? 

35.  A  ladder  40  feet  long  stands  against  a  house, 
the  foot  resting  on  the  ground,  on  a  level  with  the 
foundation  of  the  house,  and  20  feet  distant  from  it. 
How  far  up  will  it  reach? 

36.  The  floor  of  a  room  measures  16  feet  in  length, 
and  14  feet  in  width.  How  long  a  line  will  reach 
diagonally  from  corner  to  corner  ? 

37.  The  two  parts  of  a  carpenter's  square,  one  12, 
the  other  24  inches  long,  may  be  regarded  as  the  legs 
of  a  right-angled  triangle.  How  long  would  be  the 
hypotenuse  connecting  their  extremities  ? 

38.  There  is  a  room  16  feet  long,  14  feet  wide,  and 
10  feet  high.  How  long  must  a  straight  line  be, 
reaching  from  a  corner  of  the  room  at  the  bottom  to 
the  diagonal  corner  at  the  top  ? 

39.  There  is  a  room,  the  length,  breadth,  and 
height  of  which  are,  each,  10  feet.  How  far  is  it 
from  a  corner  of  the  room  at  the  bottom  to  the 
diagonal  corner  at  the  top  ? 

40.  There  is  a  room,  the  length,  breadth,  and 
height  of  which  are  equal.  The  distance  from  a 
corner  at  the  bottom  to  the  diagonal  corner  at  the 
top  is  18  feet.     What  is  the  size  of  the  *room  ? 

20 


230 


EXTRACTION    OF    THE    CUBE    ROOT. 


41.  I  have  a  cubic  block  measuring  4  inches  each 
way.     How  far  apart  are  its  diagonal  corners  ? 

42.  How  large  a  cube  can  be  cut  from  a  sphere 
which  is  1  foot  in  diameter  ? 


SECTION    XXXVIII. 


EXTRACTION  OF  THE  CUBE  ROOT. 

[See  Section  XIX.,  Part  I.] 

We  will  first  consider  those  numbers  the  cube  root 
of  which  is  expressed  by  a  single  figure.  Every  exact 
cube,  of  not  more  than  three  figures,  will  have  for  its 
root  some  number  less  than  10,  and,  consequently,  it 
will  be  expressed  by  a  single  figure.  This  root  can 
be  found  by  successive  trials. 

Examples. 

1.  What  is  the  cube  root  of  125? 

2.  What  is  the  cube  root  of  216? 

3.  What  is  the  cube  root  of  512? 

4.  What  is  the  cube  root  of  729? 

We  will  next  take  perfect  cubes,  the  root  of  which 
consists  of  two  figures. 


5.  What  is  the  cube 
root  of  4096  ? 


300 
30 


Operation. 

4096 
1000 


330 


3096 


10,   1st  part  of  the  root. 
6,  2d  part  of  the  root. 


16,  Ans. 


1800 

1080 

216 

3096 


0000 


EXTRACTION  OF  THE  CUBE  ROOT.        231 

Rule. — Place  a  period  over  the  unit  figure,  and 
another  over  that  of  thousands.  Tind  the  greatest 
cuhe  in  the  first  period,  whose  root  is  expressible  in 
tens.  Set  down  this  root  as  a  quotient  in  division. 
Find  the  cube  of  the  root,  and  subtract  it  from  the 
first  period,  and  bring  down  the  second  period  as  a 
dividend.  At  the  left  hand  of  this  set  down  three 
times  the  square  of  the  root,  and  under  this  three 
times  the  root ;  add  these  together,  for  a  trial  divisor. 
Find,  by  trial,  what  the  next  figure  of  the  root  will 
be,  and  set  it  under  the  first  part,  already  found. 
Multiply  by  this  figure  three  times  the  square  of  the 
first  part  of  the  root,  setting  the  product  under  the 
dividend.  Multiply  by  the  square  of  this  figure  three 
times  the  first  part  of  the  root,  setting  the  product 
underneath  the  other.  Under  these  set  the  cube  of 
the  root  figure  last  found.  Add  these  three  numbers 
together,  and  subtract  their  sum  from  the  dividend. 
If  the  work  be  correct,  there  will  be  no  remainder. 
Add  together  the  two  parts  of  the  root  for  the  answer. 

6.  What  is  the  cube  root  of  2744  ? 

7.  What  is  the  cube  root  of 

8.  What  is  the  cube  root  of 

9.  What  is  the  cube  root  of 

10.  What  is  the  cube  root  of 

11.  What  is  the  cube  root  of 

We  will  next  consider  the  case  where  there  are 
more  than  two  figures  in  the  root.  The  number  of 
figures  in  the  root  can  always  be  determined  by  the 
number  of  periods  placed  over  the  sum,  beginning 
with  units,  and  placing  a  period  over  every  third 
place.  If  there  are  more  than  three  periods  in  the 
cube,  regard,  first,  only  the  two  left-hand  periods, 
obtaining  the  first  and  second  figures  of  the  root,  just 
as  if  they  constituted  the  whole  root.  Then,  after 
bringing  down  the  figures  of  another  period,  add  the 


2744? 

Of  205379  ? 

3375? 

Of  5832  ? 

4913? 

Of  10648? 

9261? 

Of  15625? 

13824? 

Of  19683? 

46656  ? 

Of  39304  ? 

232        EXTRACTION  OF  THE  CUBE  ROOT. 

two  parts  of  the  root,  and  consider  their  sum  as  the 
first  part  of  the  root,  and  proceed  to  find  the  next 
part.  To  indicate  this,  you  must  annex  a  cipher  to 
the  figures  of  the  root  already  found. 

12.  What  is  the  cube  root  of  1953125? 
«      13.   What  is  the  cube  root  of  2406104? 

14.  What  is  the  cube  root  of  3796416? 

If  there  are  decimals  in  the  given  sum,  point  off 
both  ways  from  the  units'  place,  adding  ciphers,  if 
necessary,  to  the  decimal,  in  order  to  make  the  period 
complete. 

15.  What  is  the  cube  root  of  15.625? 

16.  What  is  the  cube  root  of  35.937? 

If  the  number  given  is  not  a  perfect  cube,  add 
periods  of  ciphers,  and  carry  out  the  root  in  decimals 
as  far  as  may  be  desired. 

17.  What  is  the  cube  root  of  10? 

18.  What  is  the  cube  root  of  20? 

19.  What  is  the  cube  root  of  50? 

20.  What  is  the  cube  root  of  100? 

21.  A  bushel,  even  measure,  contains  2152  solid 
inches.  What  would  be  the  inside  measure  of  a 
cubic  box  containing  12  bushels? 

22.  A  gallon,  wine  measure,  contains  231  cubic 
inches.  What  must  be  the  inside  measure  of  a  cubic 
cistern  containing  10  barrels  ? 

23.  What  would  be  the  measure  of  a  cubic  pile  of 
wood  containing  one  cord  ? 


PROPORTION.  233 

SECTION     XXXIX. 

PROPORTION. 

[See  Section  XX.,  Part  I.] 

Several  changes  that  may  be  made  in  the  terms  of  a 
proportion  are  exhibited  in  page  131.  In  continuing 
the  subject,  we  will  first  state  some  further  changes 
that  may  be  made  in  the  terms  without  destroying  the 
proportion. 

1.  Multiply  all  the  terms  by  the  same  number. 

2.  Divide  all  the  terms  by  the  same  number. 

3.  Add  the  terms  of  the  first  ratio  for  the  first  an- 
tecedent, and  the  terms  of  the  second  ratio  for  the 
second  antecedent. 

4.  Add  the  terms  of  the  first  ratio  for  the  first  con- 
sequent, and  the  terms  of  the  second  ratio  for  the 
second  consequent. 

5.  Instead  of  the  sum  of  the  terms  in  the  third  case 
above,  take  the  difference  of  the  terms. 

6.  Instead  of  the  sum  of  the  terms  in  the  fourth 
case  above,  take  the  difference  of  the  terms. 

7.  Raise  each  term  to  the  same  power,  as  second  or 
third  power. 

8.  Extract  of  each  term  the  same  root. 

The  result,  after  each  of  these  operations,  will  still 
be  a  proportion,  and  may  be  proved  to  be  so,  by  mul- 
tiplying the  extremes  together,  and  finding  the  product, 
equal  to  that  of  the  means. 

Take  the  proportion,  4  :  16  : :  9 :  36,  and  perform  on 
it  the  first  change,  using  any  number  you  please  for  a 
multiplier,  and  then  prove  the  proportion. 

Perform  on  the  same  proportion  the  second  change. 

Perform  the  third  change. 

Perform  the  fourth  change. 
20* 


234 


PROPORTION. 


Perform  the  fifth  change. 

Perform  the  sixth  change. 

Perform  the  seventh  change,  raising  to  the  second 
power. 

Perform  the  eighth  change,  extracting  the  square 
root. 

Finally,  you  may,  in  any  case,  invert  the  whole 
proportion ;  or,  invert  the  terms  of  each  ratio ;  or, 
invert  the  means  or  the  extremes. 


Practical   Questions. 

1.  If  7  lbs.  of  flour  cost  31  cents,  what  will  196 
lbs.  cost? 

As  the  smaller  quantity  is  to  the  larger  quantity,  so 
is  the  price  of  the  smaller  quantity  to  the  price  of  the 
larger. 

2.  If  3  cwt.  of  hay  cost  2  dollars,  what  will  35  cwt. 
cost  ? 

3.  If  4  qts.  of  molasses  cost  38  cents,  what  will  10 
qts.  cost? 

4.  If  a  horse  travels  19  miles  in  3  hours,  how  far 
will  he  travel  in  11  hours? 

5.  If  the  freight  of  7  cwt.  cost  2  dollars,  what  will 
the  freight  of  20  cwt.  cost  ? 

6.  If  11  dollars  buy  3  cords  of  wood,  how  many 
cords  will  50  dollars  buy  ? 

7.  If  7  bushels  of  oats  last  a  horse  2  months,  how 
long  will  23  bushels  last  him,  at  the  same  rate  ? 

8.  A  man  bought  a  horse  for  84  dollars,  and  sold 
him  for  $93.     What  did  he  gain  per  cent.? 

As  the  whole  outlay  is  to  1  dollar,  so  is  the  whole 
gain  to  the  gain  on  a  dollar. 

9.  A  merchant  buys  flour  at  $4.35  a  barrel,  and  sells 
it  for  $4.63.     What  is  his  gain  per  cent.  ? 

1.0.  A  and  B  form  a  partnership  in  trade.     A  puts  in 


PROPORTION.  235 

$500,  and  B  $300,  for  the  same  time.     They  gain 
$  180.     What  ought  each  to  share  ? 

As  the  whole  stock  is  to  each  one's  share,  so  is  the 
whole  gain  to  each  one's  gain. 

11.  C  and  D  trade  in  company.  C  puts  in  750  dol- 
lars, and  D  $450,  for  the  same  time.  They  gain  240 
dollars.     How  much  gain  ought  each  to  receive  ? 

12.  Two  men  buy  a  lot  of  wood  in  company  for 
340  dollars.  One  takes  away  42  cords  ;  the  other,  the 
remainder,  which  was  34  cords.  What  ought  each  to 
pay? 

13.  Two  men  hire  a  sheep-pasture  in  company  for 
20  dollars.  One  keeps  30  sheep  in  it  14  weeks ;  the 
other,  24  sheep,  16  weeks.     What  ought  each  to  pay  ? 

Find  how  many  weeks'  pasturing  for  a  single  sheep 
each  one  had. 

14.  Two  men  purchase  a  lot  of  standing  grass  for 
$36.50.  One  takes  34  tons  ;  the  other,  If  tons. 
What  ought  each  to  pay? 

Reduce  the  quantity  of  hay  to  fourths  of  a  ton,  and 
then  state  the  proportion. 

15.  'There  is  a  circular  piece  of  ground,  whose  di- 
ameter is  14  rods.  What  will  be  the  diameter  of  a 
circle  containing  twice  as  much  ? 

16.  There  is  a  circular  piece  of  ground  containing 
2.5  acres.  What  will  be  the  area  of  a  circle,  the 
diameter  of  which  is  3  times  as  great  ? 

17.  There  are  two  similar  triangular  fields.  The 
smaller  contains  3  acres,  the  larger  4.  The  base  of 
the  smaller  is  44  rods.  How  long  is  the  base  of  the 
larger  ? 

18.  There  are  two  similar  rectangular  fields.  The 
smaller  is  34  rods  wide,  and  60  rods  long.  The  other 
has  twice  as  great  an  area.     What  are  its  dimensions? 

19.  There  is  a  grindstone  4  feet  in  diameter.  What 
will  be  its  diameter  after  half  of  it  is  ground  off  ? 


236  PROPORTION. 

20.  There  are  two  similar  triangular  pieces  of  land. 
The  base  of  one  measures  44  rods.  The  other  piece 
has  an  area  7  times  as  large  as  the  first.  What  is  the 
length  of  its  base  ? 

21.  There  are  two  cisterns  of  the  same  shape.  One 
is  5  feet  deep.  The  other  has  a  capacity  three  times 
as  great.     How  deep  is  it  ? 

22.  If  a  ball  5  inches  in  diameter  weighs  14  lbs., 
what  will  be  the  weight  of  one  of  the  same  material 
6  inches  in  diameter  ? 

23.  What,  on  the  same  supposition,  will  be  the 
weight  of  a  ball  of  7  inches  diameter  ? 

24.  There  are  two  marble  statues  of  the  same  form, 
but  differing  in  size.  One  is  5  feet  high,  and  weighs 
740  lbs.  The  other  is  7  feet  high.  What  will  it 
weigh  ? 

25.  If  a  tree,  2£  feet  in  diameter  at  the  ground, 
contains  3  cords  of  wood,  how  much  will  there  be  in 
a  tree  of  the  same  shape,  3£  feet  in  diameter? 

26.  There  are  two  similar  stacks  of  hay.  The 
smaller  is  ll£  feet  high,  and  contains  4£  tons  of  hay. 
The  larger  is  14  feet  high.  How  much  hay  does  it 
contain,  supposing  both  to  be  of  the  same  solidity  ? 

27.  If  an  iron  field-piece,  5J-  feet  long,  weighs  1140 
lbs.,  how  many  lbs.  will  an  iron  cannon  of  the  same 
shape  weigh,  that  is  lOf  feet  long  ? 

28.  There  are  two  anchors  of  similar  form.  The 
smaller  weighs  1100  lbs.  The  larger  is  2f  times  as 
long.      What  is  its  weight  ? 

When  a  cause  and  an  effect  are  connected  together, 
the  increase  of  the  one  is  always  coifnected  with  an 
increase  of  the  other.  If  6  horses  eat  20  bushels  of 
oats,  we  may  regard  the  horses  as  the  cause,  and  the 
consumption  of  the  oats  the  effect ;  or,  if  we  please, 
we  may  regard  the  oats  as  the  cause,  and  the  support 
of  the  horses  as  the  effect.     But,  in  either  case,  an 


PROPORTION'.  237 

increase  of  one  would  require  an  increase  of  the  other. 
When  numbers  are  connected  in  this  way,  in  a  pro- 
portion, having  the  relation  of  cause  and  effect  to  each 
other,  the  proportion  is  said  to  be  direct. 

But  it  often  happens  that  quantities  are  connected 
together,  not  as  cause  and  effect,  but  as  limitations  of 
each  other ;  where  an  increase  of  one  quantity  re- 
quires a  diminution  of  the  other. 

Thus,  if  the  provisions  of  a  ship's  company  are  suf- 
ficient to  last  17  weeks,  at  the  rate  of  13  oz.  of  bread 
per  day  for  each  man,  it  is  evident  that  these  quanti- 
ties, 17  and  13,  are  not  cause  and  effect,  but  limitations 
of  each  other.  If  one  is  increased,  the  other  must  be 
diminished.  So,  if,  with  a  speed  of  11  miles  per  hour, 
a  journey  be  performed  in  31  hours,  it  is  evident  that 
an  increase  of  one  term  must  diminish  the  other. 
When  quantities  mutually  limiting  each  other  enter 
into  a  proportion,  it  is  called  indirect  proportion.  No 
special  rule,  however,  is  needed  for  the  statement  of 
such  questions;  for  you  can  always  determine,  by 
strict  attention,  whether  the  statement  you  make  is 
reasonable. 

29.  If,  with  a  speed  of  11  miles  per  hour,  a  journey 
is  performed  in  37  hours,  how  long  will  it  take  to  per- 
form the  same  journey  with  a  speed  of  15  miles  per 
hour  ? 

30.  If  a  stable-keeper  has  grain  for  the  supply  of  29 
horses  43  days,  how  long  will  the  supply  last,  if  he 
buys  6  horses  more  ? 

31.  If  a  barrel  of  flour  last  a  family  of  7  persons  6 
weeks,  how  long  will  it  last  15  persons? 

32.  If.  42  men  can  do  a  job  of  work  in  60  days, 
how  long  will  it  take  53  men  to  perform  the  same 
work  ? 

33.  A  ship-builder  employs  50  men  to  complete  a 
ship,  which  they  can  do  in  45  days.     If  7  of  the  men 


238  COMPOUND    PROPORTION. 

fail  to  engage  in  the  work,  how  long  will  it  take  the 
others  to  perform  it  ? 

34.  If  8  yds.  of  cloth,  7  qrs.  wide,  cost  54  dollars, 
what  will  be  the  cost  of  15  yds.  of  cloth,  of  the  same 
quality,  9  qrs.  wide? 

In  this  example,  the  length  of  the  two  pieces  of 
cloth  will  not  represent  the  ratio  of  their  values,  for 
they  are  of  different  widths.  The  answer  can  be 
found  by  two  statements.  First,  regarding  the  two 
pieces  as  of  the  same  width, 

8  :  15  :  :  54  :  10l£  dollars,  first  answer. 

Next,  taking  the  width  into  view, 

7  :  9  :  :  10 1£  :  130^8  dollars,  final  answer: 

If  we  examine  the  above  question,  we  shall  see 
that  in  neither  of  them  is  the  quantity  of  cloth  ex- 
pressed; but,  in  the  first  statement,  its  length,  and  in 
the  second,  its  width.  Now,  the  quantity  of  cloth  is 
expressed  by  the  length  multiplied  by  the  breadth.  In 
the  smaller  piece,  it  is  7X8  =  56  qrs.  of  a  square  yard; 
in  the  larger  piece,  it  is  15X9=135  qrs.  of  a  square 
yard.  These  numbers,  56  and  135,  express  the  quan- 
ties  of  cloth  ;  and  taking  these,  instead  of  the  dimen- 
sions, a  single  proportion  gives  us  the  answer;  — 

56:  135::54:'1302V 

As  the  question  is  first  stated,  you  observe  that,  in- 
stead of  the  numbers  which  form  the  ratio,  56  :  135, 
you  have  only  the  factors  of  those  numbers  given. 
This  is  called  a  Compound  Proportion. 

A  compound  proportion,  then,  is  one  in  which  two 
or  more  of  the  terms  of  the  simple  proportion  are  ex- 
pressed in  the  form  of  their  factors. 

Every  question  containing  a  compound  proportion 
may  be  solved  by  means  of  two  or  more  simple  pro- 
portions :  or,  it  may  be  reduced  to  one  simple  propor- 


COMPOUND    PROPORTION.  239 


tion,  as  is  seen  in  example  34,  above.  This  method, 
however,  often  requires  calculations  in  large  numbers. 
It  may  therefore  be  desirable  to  have  a  method  given 
by  which  the  process  may  be  made  less  tedious. 

The  following  rule  is  offered,  as  applicable  to  all 
cases  of  proportion,  simple  or  compound,  —  direct  or 
inverse.  It  is  short,  and  the  attention  required  in 
applying  it  will  afford  a  good  discipline  for  the  rea- 
soning powers. 

Rule  of  Proportion. 

Draw  a  horizontal  line.  Then  examine  the  condi- 
tions of  the  question,  and  consider,  in  the  case  of 
each,  whether  its  increase  would  make  the  answer 
greater  or  smaller.  If  it  would  make  it  greater,  set  it 
above  the  line ;  if  smaller,  set  it  below. 

Regard  the  numbers,  thus  set  down,  as  the  terms  of 
a  compound  fraction.  Cancel  common  factors.  Mul- 
tiply together  the  terms  that  remain,  for  the  answer. 

Example. 

35.  If  8  men  build  a  wall  36  ft.  long,  12  ft.  high, 
and  4  ft.  thick,  in  72  days,  when  the  days  are  9  hours 
long,  how  many  men  will  build  a  wall  100  ft.  long, 
10  ft.  high,  and  3  ft.  thick,  in  24  days,  when  the  days 
are  10  hours  long? 

Cancelling  like  factors  above  and  below  the  line, 
and  multiplying  the  remaining  terms, 

Operation. 

8  72  9  100  10  3 
36  12  4    24  10     =a*2  =  37*  men>  Answer- 

Explanation. 

The  question  is,  "How  many  men?"  "If  8  men 
will  build,"  &c.     Now,  if  it  took  80  men  to  build  the 


240 


PROPORTION. 


first  wall,  instead  of  8,  it  would  require  more  men  to 
build  the  second;  then  put  8  above  the  line.  "Build 
a  wall  36  ft.  long."  If  it  were  360  feet  long,  instead 
of  36,  it  would  take  fewer  men  to  build  the  second 
wall;  therefore  put  36  below  the  line.  Pursue  the 
same  reasoning  with  all  the  other  conditions. 

36.  If  15  horses  consume  40  tons  of  hay  in  30 
weeks,  how  many  horses  will  consume  56  tons  of  hay 
in  32  weeks? 

37.  If  1  dollar  gain  .06  of  a  dollar  interest,  in  12 
months,  how  much  will  740  dollars  gain  in  8  months? 

38.  If  a  crew  of  75  men  have  provisions  for  5 
months,  allowing  each  man  30  oz.  per  day,  what  must 
be  the  allowance  per  day,  to  make  the  provisions  last 
6^  months? 

39.  If  18  bricklayers,  in  12  days,  of  9  hours  each, 
build  a  wall  175  feet  long,  2  feet  thick,  and  18  feet 
high,  in  how  many  days  will  6  men,  working  10  hours 
each  day,  build  a  wall  100  feet  long,  1J  feet  thick,  and 
16  feet  high  ? 

40.  If  10  masons  lay  160  thousand  of  bricks  in  12 
days,  working  8  hours  per  day,  how  many  men  will 
lay  224  thousand  in  15  days  of  10  hours  each? 

Partnership. 

41.  Two  men  trade  in  company.  One  puts  in  1000 
dollars,  the  other  2000,  for  the  same  length  of  time. 
They  gain  600  dollars.  What  is  each  one's  share 
of  the  gain? 

It  is  evident  that  each  man's  share  ought  to  be  in 
proportion  to  the  sum  he  put  in. 

As  the  whole  investment  is  to  each  partner's  invest- 
ment, so  is  the  whole  gain  or  loss  to  each  partner's 
gain  or  loss. 

42.  Two   men   trade   in   company.     One   puts   in 


PROPORTION'. 


241 


3500  dollars ;  the  other,  4000.     They  gain  600  dol- 
lars.    What  is  each  one's  share  of  the  gain  ? 

43.  Three  men  trade  in  company.  The  first  puts 
in  3400  dollars ;  the  second,  800 ;  and  the  third,  1200 
dollars.  They  gain  475  dollars.  What  is  each  part- 
ner's share  ? 

44.  Two  men  purchase  a  ship  for  11000  dollars. 
One  pays  8000  dollars ;  the  other,  3000.  They  sell 
the  ship  for  9500  dollars.     What  is  each  one's  loss? 

45.  Two  men  trade  in  company.  One  puts  in  1000 
dollars  for  6  months ;  the  other  puts  in  1000  for  18 
months.  They  gain  600  dollars.  What  is  each  one's 
share  of  the  gain? 

Here,  though  the  money  was  equal,  it  is  evident 
the  gain  of  one  ought  to  be  three  times  as  great  as  the 
other,  because  his  money  was  in  three  times  as  long. 

Where  the  investments  are  made  for  different  times, 
each  partner's  interest  will  be  expressed  by  multiply- 
ing his  money  by  the  time  it  was  in  trade.  Then,  as 
the  sum  of  all  the  interests  is  to  each  partner's  interest, 
so  is  the  whole  gain  or  loss  to  each  partner's  gain  or 
loss. 

46.  Three  men  trade  in  company.  The  first  puts 
in  400  dollars  for  8  months;  the  second,  1100  dollars 
for  6  months;  the  third,  1000  dollars  for  7  months. 
They  gain  840  dollars.     What  is  each  man's  share? 

47.  Four  men  trade  in  company.  The  first  puts  in 
1200  for  2  years;  the  second,  1500  dollars  for  18 
months;  the  third,  600  for  8  months;  the  fourth,  900 
dollars  for  6£  months.  They  gain  1340  dollars.  What 
is  each  man's  share  ? 

General  Rule  for  Cancelling. 

In  all  operations  involving  simply  multiplication  and 
division,  set  the  multipliers  above  the  line,  in  the  form 

21  Q 


242 


PROPORTION. 


of  their  factors,  and  the  divisors  below.  Cancel  com- 
mon factors,  and  the  resulting  compound  fraction  will 
be  the  answer. 

Examples. 

48.  How  many  blocks  of  granite,  16  inches  long, 
12  inches  wide,  and  8  inches  thick,  will  it  take  to 
make  a  solid  pile  128  feet  long,  48  feet  wide,  and  36 
feet  high  ? 

Here,  the  dimensions  of  the  pile  constitute  the  fac- 
tors, which  are  to  be  multiplied  together  to  form  the 
dividend.  The  dimensions  of  a  block  are  the  factors 
which  are  to  be  multiplied  together  to  form  the  di- 
visor.    The  operation,  then,  is  as  follows:  — 

Length.  m  Width.  Height.  Blocks. 

X%X£X$$l]  12X^X12;  J2X3X  12  =  248832,  Ans. 
X$XMX$ 

49.  How  many  cubic  yards  are  there  in  a  cellar  60 
feet  long,  45  feet  wide,  and  9  feet  deep? 

50.  What  are  the  solid  contents,  in  yards,  of  a  stone 
wall  42  feet  long,  15  feet  high,  and  44  feet  thick? 

As  4J  ==  f ,  the  9  goes  with  the  multipliers,  and  the 
2  with  the  divisors  ;   thus, 

0X7X3X5X0=105  cubic  yards,  Ans. 

$X#X#X2 

In  all  cases,  Reduce  mixed  numbers  to  improper 
fractions ;  and  if  they  are  multipliers,  write  the  terms 
in  their  natural  order,  the  numerator  above,  and  the 
denominator  below,  the  line.  If  they  are  divisors,  in- 
vert them. 

51.  What  are  the  solid  contents,  in  yards,  of  a  wall 
48  feet  long,  7£  feet  high,  and  2£  feet  wide  ? 

52.  What  is  the  cost  of  digging  a  cellar  63  feet 
ong,  18  feet  wide,  and  7£  feet  deep,  at  20  cents  a 

cubic  yard  ? 


PROGRESSION.  243 

As  20  cents  =  |  of  a  dollar,  this  fraction  is  a  multi- 
plier, giving  the  answer  in  dollars.  If  20  is  used,  it 
will  give  the  answer  in  cents. 

53.  How  many  cords  of  wood  are  there  in  a  pile 
164  feet  long,  4  feet  wide,  and  12  feet  high? 

54.  How  many  cords  of  wood  are  there  in  a  pile  80 
feet  long,  32  feet  wide,  and  7  feet  high? 

55.  What  is  the  value  of  a  pile  of  wood,  at  $4.00  a 
cord,  the  dimensions  of  which  are  90  feet,  8  feet,  and 
6  feet  ? 

56.  What  is  the  cost  of  cutting  a  pile  of  wood,  at 
60  cents  a  cord,  the  dimensions  of  which  are  44  feet, 
8  feet,  and  5i  feet? 

One  eighth  of  a  cord,  or  a  pile,  1  ft.  X4  ft.X4  ft.  = 
16  feet,  is  called  a  cord  foot. 

57.  How  many  cord  feet  are  there  in  a  load  of 
wood  8  feet   long,  4£  feet  wide,  and  3^  feet  high? 

$X9X7  =  63 

- — - =7|-  cord  feet,  Ans. 

4X4X^X2=8 

58.  How  many  cord  feet  in  a  load  of  wood,  8  feet 
long,  4£  feet  high,  and  5£  feet  wide  ? 

59.  How  mauy  cord  feet  in  a  load  of  wood,  7  feet  6 
inches  long,  4  feet  3  inches  wide,  and  3  feet  4  inches 
high? 

60.  What  is  the  value  of  a  load  of  wood,  8  feet 
long,  4$  feet  wide,  and  2  feet  high,  at  $  of  a  dollar  a 
cord  foot  ? 


SECTION    XL. 

PROGRESSION. 

When  a  series  of  numbers  is  given,  each  one  of 
which  has  the  same  ratio  to  the  number  which  fol- 
lows it,  the  series  is  called  a  progression. 


244  PROGRESSION. 

Progression  is  arithmetical  or  geometrical.  Arith- 
metical progression  is  made  by  the  successive  addition 
or  subtraction  of  a  common  difference.  When  the 
common  difference  is  added  to  each  term,  in  order  to 
make  the  succeeding  one,  the  series  is  called  an 
ascending  series;  as,  1,  3,  5,  7,  9,  11,  &c. 

When  the  common  difference  is  subtracted,  the 
series  is  called  a  descending  series;  as,  11,  9,  7,  5,  3,  1. 

If  you  know  the  first  term  and  the  common  dif- 
ference of  an  arithmetrical  progression,  you  can  write 
the  whole  series,  for  to  do  this,  you  have  only  to  add 
or  subtract  the  common  difference  for  each  succeeding 
term.  If  the  whole  series  is  written  out,  it  is  evident 
you  can  find  by  inspection  any  particular  term,  as  the 
7th,  the  15th,  the  20th,  &c.  But,  if  the  series  be  a 
long  one,  this  may  be  a  very  tedious  operation. 

Suppose  the  series  given  above,  1,  3,  5,  7,  &c, 
were  continued  to  87  terms,  and  you  were  required 
to  find  what  was  the  last  term. 

By  examining  the  series,  you  will  see  that  the  2d 
term  =  the  1st  +  the  common  difference  ;  the  3d  = 
1st  +  twice  the  common  difference  ;  the  4th  =  1st +3 
times  the  common  difference;  the  5th  =  1st +  4  times 
the  common  difference ;  &c.  Any  term  whatever  equals 
the  1st  term  +  the  common  difference,  multiplied  by 
a  number  one  less  than  that  which  expresses  the  place 
of  the  term.  The  87th  term  in  the  above  series,  there- 
fore, is  1  +  2X86=173. 

1.  What  is  the  38th  term  of  the  series  1,  3,  5,  7,  &c.  ? 

2.  What  is  the  53d  term  of  the  same  series?  The 
91st  term?     The  89th  term?     The  107th  term? 

3.  In  an  arithmetical  series,  the  first  term  of  which 
is  1,  and  the  common  difference  3,  what  is  the  64th 
term?     The  75th  term?     The  81st  term? 

4.  In  the  series  1,  5,  9,  13,  &c,  what  is  the  40th 
term?     The  67th  term?     The  80th  term? 


ARITHMETICAL    PROGRESSION.  245 

5.  In  the  series  2,  4,  6,  &c,  what  is  the  45th  term  ? 
What  is  the  100th  term  ?     What  is  the  200th  term  ? 

Hence,  if  you  know  the  number  and  place  of  any 
term,  and  the  common  difference,  you  may  find  the  first 
or  any  other  term. 

6.  If  the  5th  term  of  an  arithmetical  series  is  13, 
and  the  common  difference  3,  what  is  the  1st  term? 
What  is  the  24th  term  ?     What  is  the  191st  term  ? 

7.  If  the  6th  term  of  a  series  is  77,  and  the  com- 
mon difference  15,  what  is  the  2d  term  ?  What  is  the 
14th  term? 

8.  If  the  22d  term  in  a  series  is  89,  and  the  com- 
mon difference  4,  what  is  the  10th  term  ?  What  is 
the  43d  term  ? 

By  knowing  the  number  and  the  place  of  any  two 
terms,  we  may  find  the  common  difference. 

9.  In  a  certain  series,  the  4th  term  is  10,  and  the  7th 
term  is  19.     What  is  the  common  difference  ? 

19 — 10  =  9.  Now,  this  difference,  9,  is  made  by 
the  addition  of  the  common  difference  three  times ; 
for  7 — 4  =  3.  The  common  difference,  therefore,  is 
9^3  =  3. 

10.  In  a  certain  series,  the  5th  term  is  9,  and  the 
11th  term  is  21.     What  is  the  common  difference  ? 

11.  In  a  certain  series,  the  4th  term  is  13,  and  the 
9th  term  is  33.     What  is  the  common  difference  ? 

If  we  know  the  1st  term,  the  common  difference, 
and  the  number  of  terms,  we  can  find  the  sum  of  all 
the  terms. 

12.  How  many  strokes  does  a  clock  strike  in  24 
hours,  from  noon  to  noon? 

We   might  write   down  the   series  1,  2,  3,  &c,  up 

to  12,  which  would  express  the  number  of  strokes  in 

12  hours,  from  noon  till  midnight;  we  might  write 

the  same  series  again,  for  the  time  from  midnight  till 

21* 


246  ARITHMETICAL    PROGRESSION. 

noon ;  and,  by  adding  these  numbers  together,  might 
obtain  the  answer.  But  a  much  shorter  way  may  be 
found.  To  exhibit  it,  we  will  write  the  two  series 
thus : — 

1st  series,  1  2  3  4  5  6  7  8  9101112,  from  noon  till  midnight. 
2d  series,    121110      9      8       7      6      5      4       3      2       1,    from  midnight  till  noou. 

13  13  13  13  13  13  13  13  13  13  13  13 

Sum  of  both  series  equal  to  12  times  13.  But  13 
is  the  sum  of  the  first  and  last  terms,  and  12  is  the 
number  of  terms.  Therefore,  The  sum  of  the  first 
and  last  terms,  multiplied  by  the  number  of  terms, 
gives  the  sum  of  all  the  terms  of  both  series.  Half 
this  number  will  be  the  sum  of  one  series. 

13.  What  is  the  sum  of  the  series  1,  4,  7,  to  20 
terms  ? 

First  find  the  20th  term. 

14.  What  is  the  sum  of  50  terms  of  the  series  2, 
6,  10? 

15.  A  farmer  instructed  his  boy  to  carry  fencing- 
posts  from  a  pile  to  the  holes  in  the  ground  where 
they  were  to  be  inserted,  taking  one  post  at  a  time. 
The  holes  are  12  feet  apart,  in  a  straight  line,  and  the 
pile  of  posts  30  feet  from  the  first  hole.  How  far 
must  he  travel,  in  carrying  to  their  places  100  posts  ? 

16.  If  the  hours  in  a  whole  week  were  numbered 
in  regular  progression,  and  were  struck  in  this  way  by 
the  clock,  how  many  strokes  would  the  clock  strike 
for  the  last  hour  of  the  week? 

What  would  be  the  whole  number  of  strokes  in  the 
week  ? 

If  we  know  the  first  and  last  terms  and  the  com- 
mon difference,  we  can  find  the  number  of  terms. 

17.  The  first  term  of  a  series  is  4 ;  the  last  term  is 
19 ;  the  common  difference  is  3.  What  is  the  number 
of  terms? 


ARITHMETICAL    PROGRESSION.  247 

The  difference  between  the  extremes,  19 — 4,  is  15. 
This,  you  know,  is  the  common  difference,  3,  taken  a 
certain  number  of  times  ;  15-1-3  =  5.  There  are  then 
5  additions  of  the  common  difference.  Now,  the  num- 
ber of  terms  is  1  more  than  the  number  of  times  the 
common  difference  has  been  added.  To  find  the  num- 
ber of  terms,  then, 

Find  the  difference  of  the  extremes  ;  divide  it  by  the 
common  difference  ;  increase  the  quotient  by  1  for  the 
number  of  terms. 

DESCENT   OF  FALLING   BODIES. 

18.  A  body  falling  through  the  air  falls,  in  the  1st 
second,  16.1  feet  ;*  in  the  2d  second,  48.3  feet ;  in  the 
3d  second,  80.5  feet.  How  many  feet  farther  does  it 
fall  each  second  than  it  fell  the  second  before  ? 

19.  Taking  the  answer  to  the  preceding  question  as 
the  common  difference,  and  16.1  as  the  first  term  of  a 
series,  how  far  will  a  body  fall  in  4  seconds  ? 

20.  How  far  will  a  body  fall  in  5  seconds  ? 

21.  How  far  will  a  body  fall  in  6  seconds? 

22.  A  stone,  in  falling  to  the  ground,  falls  the  last 
second  209.3  feet.  How  many  seconds  has  it  fallen, 
and  from  what  height  ? 

23.  A  stone,  in  falling  to  the  ground,  falls  the  last 
second  241.5  feet.  How  many  seconds  has  it  fallen, 
and  from  what  height  ? 

24.  If  a  stone,  dropped  into  a  well,  strikes  the  water 
in  3  seconds,  how  far  is  it  to  the  surface  of  the  water  ? 

*  This  is  a  more  exact  statement  than  that  made  in  Part  I.  (See 
Olmsted's  Natural  Philosophy.)  It  should  be  remarked,  also,  that 
no  allowance,  in  these  examples,  is  made  for  the  resistance  of  the 
atmosphere,  which  always  diminishes  the  speed  somewhat,  and  be- 
comes greater  and  greater  as  the  speed  increases. 


248  GEOMETRICAL    PROGRESSION. 

SECTION    XLI. 

GEOMETRICAL   PROGRESSION. 

A  series  of  numbers,  such  that  each  is  the  same  part 
or  the  same  multiple  of  the  number  that  follows  it,  is 
called  a  geometrical  series.  The  ascending  series  1, 
3,  9,  27,  is  of  this  kind,  for  each  term  is  one  third  of 
that  which  succeeds  it.  So,  in  the  descending  series 
64,  16,  4,  1,  each  term  is  4  times  the  following  term. 

The  number  obtained  by  dividing  any  term  by  the 
term  before  it,  is  called  the  ratio  of  the  progression. 
Thus,  in  the  first  of  the  above  examples,  the  ratio  is 
3  ;  in  the  second  example,  it  is  £. 

Let  us  take  the  series  2,  6,  18,  54,  and  observe  by 
what  law  it  is  formed.  The  ratio  is  3  ;  the  first  term, 
2.  The  second  term  is  2X3,  or  the  first  term  X  the 
ratio.  The  third  term  is  2X32,  or  the  first  term  X  the 
second  power  of  the  ratio.  The  fourth  term  is  2X33, 
or  the  first  term  X  the  third  power  of  the  ratio. 

Thus  each  term  consists  of  the  first  term  multiplied 
by  the  ratio  raised  to  a  power  whose  index  is  one  less 
than  the  number  expressing  the  place  of  the  term. 

1.  What  is  the  7th  term  in  the  series  1.,  4,  16,  &c.  ?4 

2.  What  is  the  10th  term  in  the  series  3,  6,  12,  &c.  ? 

3.  A  glazier  agrees  to  insert  a  window  of  16  lights 
for  what  the  last  light  would  come  to,  allowing  1  cent 
for  the  first  light,  2  for  the  second,  and  so  on.  What 
will  the  window  cost  ? 

4.  If,  in  the  year  1850,  the  population  of  the  United 
States  shall  be  20000000,  and  if  it  shall  thenceforward 
double  once  in  every  thirty  years,  what  will  be  the 
population  in  1970? 


GEOMETRICAL    PROGRESSION.  249 

To  obtain  the  sum  of  the  terms,  when  the  first  and 
last  terms  are  given,  and  the  ratio, 

Take  the  following  question: — What  is  the  sum  of 
a  series  whose  first  term  is  1,  last  term  64,  and  ratio  4? 

We  can  write  the  series  in  full,  and  then  obtain  the 
•  answer  by  adding  the  terms  together;  thus,  1  +  4+16 
+  64=85.  In  this  way  we  might  obtain  the  sum  of 
any  given  series ;  but  the  operation  becomes  tedious  if 
the  series  be  a  long  one,  and  hence  a  shorter  method 
is  devised. 

What  is  the  sum  of  a  series  whose  first  term  is  2, 
last  term  162,  and  ratio  3  ? 

To  show  how  a  general  rule  is  obtained,  we  will 
write  this  series  in  full ; 

thus,  2,  2X3,  2X32,  2X33,  2X3*  ; 
or,       2,      6    ,     18    ,     54    ,    162     . 

If,  now,  we  multiply  this  series  by  the  ratio,  and  set 
the  terms  of  the  product  one  degree  to  the  right,  over 
the  terms  of  the  series,  we  shall  have  this  ; 

6  +18  +  54 +162  +  486  —  3  S.  or  3  times  the  sum. 
2+6+18  +  54+162  =S.  or  once  the  sum. 

If,  now,  we  subtract  the  lower  line  of  terms  from 
the  upper,  the  remainder,  it  is  clear,  will  be  twice  the 
whole  sum  of  the  series  j  for  it  will  be  once  the  sum 
subtracted  from  3  times  the  sum.  But,  in  this  sub- 
traction, all  the  intermediate  terms  cancel  each  other, 
leaving  only  the  first  term,  2,  to  be  subtracted  from 
the  last  term,  486.  Now,  this  last  term,  486,  was 
obtained  by  multiplying  the  last  term  given  in  the 
question  by  the  ratio.  Therefore  we  might  have 
saved  all  our  work,  and  simply  multiplied  the  last 
term  by  the  ratio,  and  subtracted  the  first  term  from 
the  product.  This  would  have  given  us,  at  once, 
twice  the  sum  of  the  series,  or,  in  general  terms,  the 
series  multiplied  by  a  number  one  less  than  the  ratio. 
Hence  we  have  the  following 


250 


MENSURATION    OF    SURFACES. 


Rule.  —  Multiply  the  last  term  by  the  ratio,  sub- 
tract the  first  term  from  this  product,  and  divide  the 
remainder  by  the  ratio  diminished  by  one. 

5.  A  gentleman  promises  his  son.  11  years  old,  one 
mill  when  he  shall  be  12  years  old,  and,  on  each  suc- 
ceeding birthday,  till  he  is  21  years  old,  ten  times  as 
much  as  on  the  preceding  birthday.  What  will  the 
son's  fortune  be,  without  interest,  when  he  is  21  years 
old? 

6.  What  is  the  sum  of  14  terms  of  the  series  2,  6, 
18,  &c? 

First  obtain  the  last  term. 

7.  What  is  the  sum  of  16  terms  of  the  series  5,  10, 
20,  &c? 

8.  What  is  the  sum  of  18  terms  of  the  series  1,  2, 
4,  &c.  ? 

9.  A  builder  offers  to  build  a  church  for  16000  dol- 
lars ;  or,  if  the  people  prefer  it,  he  will  take  for  his 
pay  1  cent  for  the  first  pew,  2  cents  for  the  second,  4 
for  the  third,  8  for  the  fourth,  &c,  till  he  shall  have 
received  pay  for  forty  pews.  What  would  the  meet- 
ing-house have  cost  on  these  last-named  terms? 


SECTION    XLII. 

MENSURATION   OF   SURFACES. 

\ 

For  the  mensuration  of  the  triangle  and  the  paral- 
lelogram, when  the  base  and  height  are  known,  see 
Section  XV1L,  Part  I. 

To  find  the  area  of  an  equilateral  triangle,  when  the 
sides  only  are  known, 

Square  one  side;  multiply  that  product  by  the 
decimal  .433. 


MENSURATION  OF  SURFACES.  251 

To  find  the  circumference  of  a  circle,  when  the 
diameter  is  known, 

Multiply  the  diameter  by  3.1416. 

1.  What  is  the  circumference  of  a  circle  the  diam- 
eter of  which  is  36  feet  ? 

2.  What  is  the  circumference  of  a  circular  race- 
course whose  diameter  is  l£  mile  ? 

3.  What  is  the  circumference  of  a  wheel  the  diam- 
eter of  which  is  24  feet  6  inches  ? 

4.  What  is  the  circumference  of  the  earth  on  the 
line  of  the  equator,  its  diameter  being  7925.65  miles? 

To  find  the  diameter  of  a  circle,  when  the  circum- 
ference is  known, 

Divide  the  circumference  by  3.1416. 

5.  How  far  is  it  across  a  circular  pond,  the  circum- 
ference of  which  is  231  rods  ? 

To  find  the  area  of  a  circle,  when  the  diameter  and 
circumference  are  known, 

Multiply  the  circumference  by  one  fourth  of  the 
diameter ;  or,  multiply  the  square  of  the  diameter  by 
the  decimal  .7854. 

6.  What  is  the  area  of  a  circle  whose  diameter  is 
34  rods  ? 

7.  What  is  the  area  of  a  circle  whose  diameter  is 
24  feet  ? 

When  a  circle  is  given,  to  find  a  square  which  shall 
have  an  equal  area, 

Find  the  area,  of  the  circle,  extract  the  square  root, 
which  will  be  one  side  of  the  square. 

8.  There  is  a  circular  piece  of  land,  40  rods  in  di- 
ameter. What  will  be  the  side  of  a  square  of  equal 
area? 

9.  There  is  a  circular  green,  containing  8  acres. 
What  will  be  the  side  of  a  square  of  equal  area  ? 


252 


MENSURATION    OF     SOLIDS. 


10.  There  is  a  circle  35  rods  in  diameter,  and  a 
square  31  rods.  Which  is  the  greater ;  and  how 
much? 


SECTION    XLIII. 

MENSURATION    OF   SOLIDS. 

A  plane  solid  is  bounded  by  flat  surfaces ;  a  round 
solid  is  bounded  by  curved  surfaces. 

To  find  the  surface  of  a  solid  bounded  by  plane 
surfaces, 

Find  the  area  of  each  plane  surface,  and  add  the 
sums  together  for  the  whole  surface. 

A  prism  is  a  solid,  whose  ends  are  any  equal,  paral- 
lel, and  similar  rectilineal  figures,  and  whose  sides  are 
parallelograms. 

To  find  the  solidity  of  a  prism, 

Multiply  the  area  of  the  base  by  the  height. 

1.  What  is  the  solidity  of  a  prism  whose  ends  are 
equilateral  triangles,  14  inches  on  a  side,  and  whose 
height  is  8  feet  ? 

A  cylinder  is  a  round  solid,  whose 
ends  are  equal  and  parallel  circles. 

To  find  the  solidity  —  the  same  rule 
as  for  a  prism. 

2.  What  is  the  solid  contents  of  a  cylinder  whose 
ends  are  circles  18  inches  in  diameter,  and  whose 
height  is  12  feet  ? 

A  regular  pyramid  is  a  solid  whose  sides  are  equal 
and  similar  triangles,  meeting  in  a  point  at  the  top. 
The  slant  height  is  the  distance  from  the  point,  at  the 
top,  to  the  middle  of  the  base  of  one  of  the  triangles. 


MKXSURATION     OF     SOLIDS.  253 

To  find  the  solid  contents  of  a  pyramid, 
Multiply  the  area  of  the  base  by  one  third  the  per- 
pendicular height. 

3.  What  is  the  solid  contents  of  a  four-sided  pyra- 
mid, whose  base  measures  40  feet  on  a  side,  and  whose 
height  is  42  feet  ?  f 

A  cone  is  a  round  solid,  standing  on  a 
circular  base,  and  terminating  in  a  point 
at  the  top. 

To  find  the  solidity  of  a  cone  —  the 
same  rule  as  for  a  pyramid. 

4.  What  is  the  solidity  of  a  cone  whose 
base  is  13  feet  in  diameter,  and  whose 
height  is  22  feet  ? 

5.  What  is  the  solidity  of  a  cone  whose  base  is  4 
feet  in  diameter,  and  height  18  feet  ? 

To  find  the  surface  of  a  sphere, 

Multiply  the  diameter  by  the  circumference. 

6.  How  many  square  miles  of  surface  has  the  earth, 
regarding  it  as  a  sphere  the  diameter  of  which  is 
7925.65  miles? 

To  find  the  solidity  of  a  sphere, 
Multiply  the  surface  by  £  of  the  diameter  ;  or,  mul- 
tiply the  cube  of  the  diameter  by  the  decimal  .5236. 

To  find  the  measure  of  a  sphere,  when  the  solidity 
is  given, 

Divide  the  solidity  by  the  decimal  .5236  ;  extract  the 
cube  root  of  the  quotient,  which  will  be  the  diameter  of 
the  sphere. 

7.  What  is  the  solid  contents  of  a  sphere  the  diam- 
eter of  which  is  14  inches  ? 

8.  What  will  be  the  solidity  of  the  largest  sphere 
that  can  be  cut  from  a  cubic  block  1  foot  on  a  side  ? 

22 


254       MISCELLANEOUS    THEOREMS    AND    QUESTIONS. 


SECTION   XLIV 


MISCELLANEOUS  THEOREMS  AND  QUESTIONS. 


Given  the  sum  and  the  difference  of  two  numbers, 
to  find  the  larger  and  the  smaller  numbers, 

Add  half  the  difference  to  half  the  sum,  for  the 
larger  ;  subtract  half  the  difference  from  half  the  sum, 
for  the  smaller. 

1.  The  sum  of  two  numbers  is  140,  the  difference 
32.  What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  572,  the  difference 
94.  What  are  the  numbers  ? 

3.  The  sum  of  two  numbers  is  187,  the  difference 
44.  What  are  the  numbers  ? 

4.  The  sum  of  two  numbers  is  190,  the  difference 
57.  What  are  the  numbers? 

Given  the  sum  and  the  product  of  two  numbers,  to 
find  the  larger  and  the  smaller  number. 

5.  There  are  two  numbers  ;  their  sum  is  80,  and 
their  product  1551.     What  are  the  numbers? 

The  theorem  on  which  the  solution  of  this  question 
depends,  is  this: — If  a  number  be  divided  into  two 
equal  parts,  and  also  into  two  unequal  parts,  the 
product  of  the  two  equal  parts,  that  is,  the  square  of 
half  the  number,  will  equal  the  product  of  the  two 
unequal  parts,  plus  the  square  of  the  difference  between 
one  of  the  equal  and  one  of  the  unequal  parts. 

Take  16:  divide  it  equally,  8  +  8,  and  unequally, 
9  +  7;  the  difference  between  an  equal  and  an  unequal 
part,  1;  82  =  64:  9X7  =  63  ;  loss,  1,  which  is  the 
square  of  the  difference. 

Divide  unequally  into  10  +  6;  difference, 2;  82  =  64: 


MISCELLANEOUS    THEOREMS    AND    QUESTIONS.       255 

10X6  =  60;  loss,  4,  which  is  the  square  of  the  dif- 
ference. 

Divide  unequally  into  11  +  5;  difference,  3  ;  82  = 
64:11X5  =  55;  loss,  9,  which  is  the  square  of  the 
difference. 

Hence,  to  solve  the  question,  —  Subtract  the  product 
of  the  unequal  parts  from  the  square  of  half  the  num- 
ber ;  find  the  square  root  of  the  difference ;  add  it  to 
half  the  number  for  the  greater,  subtract  it  from  half 
the  number  for  the  less. 

6.  There  are  two  numbers  the  sum  of  which  is 
100.  Their  product  is  2419.  What  are  the  num- 
bers? 

7.  There  is  a  rectangular  piece   of  land,  the  two. 
contiguous  boundaries  of  which  measure  together  120 
rods.     The    area  of  the   piece   is  2975  square  rods. 
What  is  its  length  ?     What  is  its  width  ? 

8.  A  rectangular  piece  of  land  is  surrounded  by  480 
rods  offence.  The  area  is  13104  square  rods.  What 
is  its  length  and  breadth  ? 

Given  the  sum  of  two  numbers,  and  the  difference 
of  their  squares,  to  find  the  greater  and  the  less 
number, 

The  product  of  the  sum  and  the  difference  of  two 
numbers  is  equal  to  the  difference  of  their  squares. 

Take  the  two  numbers  6  and  9;  their  sum  is  15; 
their  difference,  3;  15X3  =  45.  The  square  of  6  is 
36;  the  square  of  9  is  81 ;  81  —  36  =  45. 

Hence,  if  we  divide  the  difference  of  the  squares  by 
the  sum,  the  quotient  will  be  the  difference,  and  from 
this  we  may  find  the  greater  and  the  less. 

9.  There  is  a  triangle,  the  hypotenuse  and  one  leg 
of  which  measure  together  90  feet ;  the  other  side 
measures  40  feet.  What  are  the  lengths  of  the  two 
first-named  sides  respectively  ? 


£56  MISC  KLLANEOIS    QUESTIONS. 

10.  There  is  a  triangle  ;  the  hypotenuse  and  base 
measure  together  120  feet ;  the  perpendicular  measures 
64  feet.  What  is  the  length  of  each  of  the  first- 
named  sides? 

This  principle  enables  you  to  multiply  readily  any 
two  numbers,  one  of  which  exceeds  a  certain  number 
of  tens  by  as  many  units  as  the  other  falls  short  of 
it ;  as,  63X57.  The  first  exceeds  60  by  3 ;  the  second 
falls  short  of  it  by  3. 

Square  the  tens  —  3600;  subtract  from  this  the 
square  of  the  units  —  9.    3591,  answer. 

11.  Multiply  64X56.     3600  — 16  =  3584,  answer. 

12.  Multiply  82X78;  47X33;  92X88. 

Theorem  of  Parallel  Sections. 

If  a  line  is  drawn  in  a  triangle,  parallel  to  one  of 
the  sides,  and  meeting  the  other  two  sides,  it  divides 
those  sides  proportionally  ;  and  the  small  triangle  cut 
off  is  similar  to  the  whole  undivided  triangle. 

If  a  plane  pass  through  a  pyramid  or  cone,  parallel 
to  the  base,  it  divides  all  the  lines  it  meets  proportion- 
ally; and  the  small  solid  cut  off  at  the  top  is  similar 
to  the  whole  undivided  solid. 

13.  There  is  a  triangular  field,  containing  7  acres. 
A  line  is  drawn  through  it,  parallel  to  one  side,  cutting 
the  other  two  sides  §  of  the  distance  from  the  apex  to 
the  base.     How  much  land  does  it  cut  off? 

14.  There  is  a  board,  in  the  form  of  a  right-angled 
triangle,  8  feet  in  perpendicular  height.  How  far 
from  the  top  must  a  line  be  drawn  parallel  to  the  base 
to  cut  off  f  of  the  board  ? 

15.  A  man  had  a  field  of  3  acres,  in  the  form  of  a 
right-angled  triangle,  with  the  base  equal  to  the  per- 
pendicular. He  sells  one  acre,  to  be  cut  off  by  a  line 
running  parallel  to  the  base.     He  then  sells  another 


MISCELLANEOUS    QUESTIONS.  257 

acre,  to  be  cut  off  by  another  line  parallel  to  the  base. 
How  far  from  the  base  must  the  first  line  be  ?  How 
far  from  the  base  must  the  second  line  be  ? 

16.  There  was  a  cone  20  feet  high ;  but,  the  upper 
part  being  defective,  11  feet  in  height  of  the  top  was 
taken  down.  How  much  of  the  cone  has  been  re- 
moved? 

17.  There  was  a  square  pyramid,  the  base  of  which 
measured  48  feet  on  a  side.  When  it  was  partly  com- 
pleted, its  slant  height,  measuring  from  the  middle  of 
a  side  at  the  bottom  to  the  middle  of  the  same  side 
at  the  top,  was  40  feet,  and  the  width  of  a  side  at 
the  top  was  12  feet.  How  high  was  the  apex  of  the 
pyramid  when  completed?  And  what  part  of  the 
pyramid  remained  to  be  built  ? 

18.  In  a  right-angled  triangle,  whose  base  and  per- 
pendicular are  equal,  what  is  the  ratio  of  the  square  of 
the  hypotenuse  to  the  square  of  the  base  ?  What  is 
the  ratio  of  the  hypotenuse  to  the  base  ? 

19.  If  a  man  travel  on  Monday  6  miles  due  north, 
and  on  Tuesday  8  miles  due  east,  how  far  is  he  then 
from  where  he  set  out  on  Monday  ? 

If,  on  Wednesday,  he  travels  12  miles  due  south, 
how  far  will  he  then  be  from  where  he  was  on  Mon- 
day morning?  How  far  from  where  he  was  on  Mon- 
day night? 

20.  In  repairing  a  meeting-house,  it  was  thought 
desirable  to  alter  the  form  of  the  posts,  which  were 
one  foot  square.  It  was  proposed  to  cut  away  the 
corners,  so  as  to  make  them  regular  eight-sided  prisms. 
How  wide  must  each  face  be,  so  as  to  have  all  the 
eight  faces  of  exactly  the  same  width  ? 

21.  Sound  moves  through  the  air  at  the  rate  of 
1090  feet  in  a  second.  How  long  would  it  be  in 
passing  100  miles  ? 

22.  At  the  above  rate,  how  long  would  it  require 
for  a  wave  of  sound  to  compass  one  half  the  circuit  of 

22*  R 


258  SPECIFIC    GRAVITY. 

the  globe,  on  the  line  of  the  equator,  the  circumfer- 
ence being  24899  miles? 

23.  Two  men  purchase,  in  equal  shares,  a  stick  of 
hewn  timber,  40  feet  long,  2  feet  square  at  the  larger 
end,  and  1  foot  square  at  the  smaller  end.  How  far 
from  the  larger  end  shall  they  cut  it  in  two,  so  that 
each  may  have  exactly  one  half? 

24.  A  surveyor,  in  laying  out  a  lot  of  land,  first  runs 
a  line  due  north,  to  a  certain  tree.  From  the  tree  he 
runs  between  south  and  west  till  he  comes  to  a  point 
due  west  from  the  place  he  started  from.  The  whole 
of  these  two  lines  is  212  rods;  but  those  who  measured 
it  neglected  to  note  how  far  the  tree  was  from  the 
starting  point.  On  measuring  a  third  line,  connecting 
the  extremities  of  the  two  first  lines,  they  find  it  98 
rods.     How  many  acres  does  the  triangle  contain  ? 

Specific  Gravity. 

The  specific  gravity  of  a  body  is  its  weight  com- 
pared with  the  weight  of  an  equal  bulk  of  water.  To 
find  the  specific  gravity  of  a  body  heavier  than  water, 

Weigh  the  body  in  water,  and  out  of  water,  and 
fi,nd  the  difference  in  the  weight.  Then,  as  the  dif- 
ference in  the  weight  is  to  the  weight  out  of  water,  so 
is  1  to  the  specific  gravity. 

The  weight  of  a  cubic  foot  of  water  is  62J-  lbs. 
avoirdupois.  The  specific  gravity  of  the  most  im- 
portant of  the  metals  is  as  follows :  — 

Iron,      7.78;         Tin,  7.2;  Copper,    8.895; 

Lead,  11.325;      Mercury,    13.568;       Silver,   10.51. 
Gold,  19.257;      Platinum,  21.25. 

From  the  above  table,  we  may  find  the  weight  of 
any  mass  of  one  of  the  above  metals  the  magnitude 
of  which  is  known. 

25.  What  is  the  weight  of  a  cubic  foot  of  iron  ? 


MECHANICAL    POWERS.  259 

26.  What  is  the  weight  of  an  iron  ball  six  inches  in 
diameter  ? 

27.  What  is  the  diameter  of  a  24  lb.  cannon  ball  ? 

28.  What  is  the  diameter  of  a  48  lb.  cannon  ball  ? 

29.  What  is  the  weight  of  a  cannon  ball  one  foot  in 
diameter  ? 

30.  If  the  column  of  mercury  in  the  barometer  be 
29J-  inches  high,  what  would  be  the  weight  of  a  col- 
umn of  mercury  of  that  height  one  inch  square  ? 

31.  As  the  weight  of  mercury  in  the  barometer 
equals  the  weight  of  the  atmosphere  on  the  same  base, 
what  is  the  pressure  of  the  atmosphere  on  a  square  foot, 
when  the  mercury  in  the  barometer  is  29  inches  high? 

The  height  to  which  water  will  rise  in  a  suction 
pump,  and  the  height  of  the  mercury  in  the  barometer, 
are  in  inverse  proportion  to  the  specific  weight  of 
those  two  bodies ;  that  is,  the  water  is  as  much  higher 
than  the  mercury,  as  mercury  is  heavier  than  water. 

32.  How  high  will  water  rise  in  a  suction  pump, 
when  the  mercury  in  the  barometer  is  291  inches 
high? 

33.  What  is  the  weight  of  a  copper  prism,  its  base 
being  an  equilateral  triangle,  3  inches  on  a  side,  and 
its  height  15  inches? 

Mechanical  Powers. 

The  object  to  be  gained  by  the  application  of  me- 
chanical powers,  is  to  overcome  a  large  weight  or 
resistance  by  means  of  a  comparatively  small  power. 

In  doing  this,  however,  the  power  must  move 
through  a  space  as  much  larger  than  the  space 
which  the  weight  moves  through,  as  the  weight  is 
heavier  than  the  power ; 

Or,  the  distance  X  tv  eight  of  the  power  =  distance 
X  weight  of  the  weight  or  mass  to  be  moved. 


260  THE    LEVER. 

This  is  the  great  law  of  mechanical  powers,  and 
applies  to  them  all,  without  exception.  In  the  practi- 
cal application  of  them,  a  certain  allowance  must  be 
made  on  account  of  the  friction  in  the  machine.  The 
amount  of  friction  differs  in  different  powers.  No  ac- 
count of  this  will  be  taken  in  the  examples  which 
follow,  unless  it  is  particularly  mentioned;  nor  will 
any  difference  be  made  between  the  power  when  in 
motion,  and  when  in  equilibrium,  or  at  rest. 

The  Lever. 

The  lever  is  a  straight  bar,  used  to  support  or  raise 
heavy  weights.  It  is  supported  by  a  prop  or  fulcrum, 
placed  near  the  weight ;  and  the  power  is  applied  at 
the  other  end  of  the  lever.  The  distance  from  the 
fulcrum  to  the  weight,  is  called  the  shorter  arm,  the 
distance  from  the  fulcrum  to  the  power,  the  longer 
arm,  of  the  lever.  If  the  lever  were  to  turn  over 
the  fulcrum  as  a  centre,  the  longer  arm  would  de- 
scribe a  larger  circle,  and  the  shorter  arm  would  de- 
scribe a  smaller  circle.  The  circumferences  of  these 
two  circles,  or  an  arc  of  the  same  number  of  degrees 
in  both,  would  be  the  distances  passed  through  by  the 
power  and  the  weight  respectively.  But  we  may  take 
the  arms  themselves  as  representing  these  distances, 
for  they  are  the  radii  of  the  two  circles;  and  the  radii 
of  different  circles  have  the  same  ratio  to  each  other 
as  the  circumferences. 

We  have,  therefore,  this  proportion:  — 

The  longer  arm  is  to  the  shorter  arm  as  the  weight 
to  the  power.  Or,  let  I-  a.  stand  for  the  longer  arm  ; 
s-  a.,  for  the  shorter ;  w.,  for  the  weight ;  and  p., 
for  the  power. 

1.  a.  :  s.  a.  :  :  w.  :  p. ;  and  any  change  admissi- 
ble in  the  terms  of  a  proportion,  may  be  made  in 
these  terms. 


THE    WHEEL    AND    AXLE.  261 

34.  If  a  lever  10  feet  long  have  its  fulcrum  one  foot 
from  the  weight,  how  great  must  the  power  be,  to 
raise  a  weight  of  1640  lbs.  ? 

35.  If  a  lever  10  feet  long  have  its  fulcrum  18 
inches  from  the  weight,  how  great  a  weight  will, 
be  raised  by  a  power  of  160  lbs.  ? 

36.  A  lever  18  feet  long  rests  on  a  fulcrum  2  feet 
from  the  end.  How  large  a  weight  can  two  men 
raise,  (one  weighing  164  lbs.,  the  other  172  lbs.,)  by- 
applying  their  weight  at  the  longer  arm  ? 

37.  If  a  lever  7£  feet  long  rest  on  a  fulcrum  15 
inches  from  the  end,  how  heavy  must  the  power  be  to 
support  a  ton  gross  weight  ? 

38.  If  the  weight  be  3600  lbs.,  and  the  power  140 
lbs.,  how  far  from  the  weight  must  the  fulcrum  be 
placed,  under  a  lever  12  feet  long,  so  as  to  have  the 
weight  and  power  balance? 

39.  If  the  weight  be  6480  lbs.,  the  power  312,  and 
the  lever  16£  feet  long,  how  far  from  the  weight  must 
the  fulcrum  be,  to  have  the  weight  and  power  bal- 
ance ? 

40.  In  a  certain  machine,  it  is  necessary  to  adjust  a 
lever  3  feet  long,  so  that  a  power  of  l±  lbs.  shall  bal- 
ance 13£  lbs.  How  far  from  the  weight  must  the  ful- 
crum be  placed? 

The  Wheel  and  Axle. 

In  this  case,  the  power  is  applied  at  the  circumfer- 
ence of  the  wheel,  and  the  weight  is  drawn  up  by  a 
rope  passing  round  the  axle,  which  is  a  smaller  wheel. 
The  principle,  therefore,  is  the  same  as  in  the  lever. 
The  semi-diameter  of  the  wheel  is  the  longer  arm  ; 
the  semi-diameter  of  the  axle,  the  shorter  arm. 

41.  In  a  grocery  store,  the  wheel  and  axle  used  in 
raising  heavy  articles^  are  of  the  following  dimensions, 


262 


THE    WHEEL    AND    AXLE.       THE    SCREW. 


viz. :  the  wheel  5  feet  in  diameter,  the  axle  7  inches 
in  diameter.  What  power  must  be  applied  to  the  rope 
passing  over  the  wheel,  to  balance  a  barrel  of  flour, 
weighing  205  lbs.,  suspended  by  a  rope  passing  over 
the  axle  ? 

42.  With  the  same  wheel  and  axle,  what  power 
will  raise  a  box  of  sugar,  weighing  431  lbs.,  adding  £ 
to  the  power,  to  overcome  the  friction  ? 

43.  In  digging  a  well,  the  wheel  employed  in  rais- 
ing stones  and  earth,  is  6  feet  in  diameter ;  the  axle, 
6£  inches  in  diameter.  What  power  will  raise  a  rock 
weighing  640  lbs.,  adding  £  to  the  power,  to  overcome 
the  friction  ? 

44.  If  a  wheel  is  14  feet  in  diameter,  what  must  be 
the  diameter  of  the  axle,  in  order  that  a  power  of  140 
lbs.  may  balance  5760  lbs.  ? 

45.  If  an  axle  is  16}  inches  in  diameter,  what  must 
be  the  diameter  of  the  wheel,  in  order  that  a  power  of 
56  lbs.  may  balance  a  weight  of  1344  lbs.  ? 

The  Screw. 

In  this  case,  the  distance  passed  through  by  the 
power  in  one  revolution,  is  equal  to  the  circumference 
of  the  circle  described  by  the  lever  which  turns  the 
screw.  The  distance  passed  by  the  weight  is  the  dis- 
tance between  two  threads  of  the  screw,  measured  in 
the  direction  of  its  axis. 

In  the  practical  application  of  this  power,  a  large 
allowance  must  be  made  to  compensate  for  the  friction. 

46.  If  the  lever  of  a  screw  is  11  feet  in  length,  and 
the  distance  of  the  threads  1|  inches,  what  power  will 
raise  a  weight  of  6431  lbs.,  making  no  allowance  for 
friction  ? 

47.  With  the  same  conditions  as  in  the  last  exam- 
ple, what  weight  will  be  raised  by  a  power  of  124 
lbs.? 


STRENGTH    OF    BEAxMS    TO    RESIST    FRACTURE.       263 

48.  What  must  be  the  length  of  the  lever  of  a 
screw,  the  threads  of  which  are  1  inch  asunder,  in 
order  that  a  power  of  3  lbs.  may  balance  a  weight  of 
1640  lbs.,  making  no  allowance  for  friction? 

49.  How  far  asunder  must  the  threads  of  a  screw 
be,  so  that,  with  a  lever  of  8£  feet  in  length,  26  lbs. 
will  balance  6590  lbs. 

Strength  of  Beams  to  resist  Fracture. 

[See  Section  XX.,  Part  I.] 

In  addition  to  the  principles  that  have  already  been 
stated  in  estimating  the  strength  of  timbers,  the  fol- 
lowing are  among  the  most  important.  It  is  under- 
stood, in  all  cases,  when  timbers  are  compared,  that 
they  are  of  the  same  wood,  and  equally  good  in 
quality. 

When  the  depth  of  two  beams  is  the  same,  and  the 
thickness  the  same,  the  strength  is  inversely  as  the 
length. 

50.  There  are  two  beams,  of  the  same  depth  and 
thickness ;  one,  18  feet  in  length ;  the  other,  13. 
The  longer  beam  will  sustain  a  weight  of  68  cwt. 
What  weight  will  the  shorter  beam  sustain  ? 

51.  Two  beams,  of  the  same  size,  measure  in  length 
22  and  17£  feet.  The  shorter  beam  will  sustain  76 
cwt.     How  much  will  the  longer  beam  sustain  ? 

52.  Two  beams,  of  equal  thickness,  have  a  depth 
of  14  and  16  inches  respectively.  The  deeper  beam 
is  20  feet  long,  and  will  sustain  84  cwt.  The  other 
is  17  feet  in  length.     What  weight  will  it  sustain  ? 

First  take  into  view  the  length;  then,  in  a  second 
proportion,  the  depth. 

53.  If  a  beam,  25  feet  in  length  and  9  inches  in 
depth,  will  sustain  a  weight  of  12  cwt..  what  weight 


264   STRENGTH  OF  BEAMS  TO  RESIST  FRACTURE. 

will  be  sustained  by  a  beam  of  the  same  thickness  18 
feet  long,  and  10  inches  in  depth  ? 

When  beams  are  of  the  same  length  and  depth,  the 
strength  varies  directly  as  the  width. 

54.  There  are  two  beams  of  equal  length  and 
depth  ;  one,  9  inches  in  width ;  the  other,  7£  inches. 
The  wider  beam  will  sustain  47  cwt.  What  weight 
will  the  narrower  beam  sustain? 

55.  There  are  two  beams  of  equal  depth.  One 
measures  20  feet  in  length,  and  11  inches  in  width, 
and  will  sustain  94  cwt.  The  other  beam  is  14  feet 
in  length,  and  10  inches  in  width.  What  weight 
will  it  sustain  ? 

56.  There  are  two  beams,  of  the  same  width.  One 
measures  16  feet  in  length,  and  10  inches  in  depth, 
and  will  sustain  66  cwt.  The  other  is  18  feet  long, 
and  12  inches  in  depth.     What  weight  will  it  sustain  ? 

It  is  sometimes  desirable  to  know  how  the  strength 
of  a  beam  will  vary  by  removing  the  point  on  which 
the  pressure  is  made,  as  in  the  following  example :  — 

57.  A  beam  20  feet  in  length  will  sustain,  at  its 
centre,  a  weight  of  44  cwt.  What  weight  will  it  sus- 
tain applied  7  feet  from  one  end  ? 

The  following  formula  will  give  the  variation  in  the 
strength :  — 

As  the  product  of  the  two  unequal  sections  of  the 
beam  (in  this  case,  13X7)  is  to  the  square  of  half  the 
length,  so  is  the  weight  which  the  beam  will  sustain 
at  the  centre  to  the  weight  it  will  sustain  at  the  other 
given  point. 

58.  A  beam  24  feet  in  length  will  sustain  at  its 
centre  56  cwt.  What  weight  will  it  sustain  at  the 
distance  of  9  feet  from  one  end  ? 


STIFFNESS    OF    BEAMS    TO    RESIST    FLEXURE, 


265 


59.  A  beam  28  feet  in  length  will  sustain  at  its 
centre  33  cwt.  What  weight  will  a  beam  of  the  same 
width  and  length,  and  of  f  the  depth  of  the  former, 
sustain  at  the  distance  of  10  feet  from  the  end  ? 


Stiffness  of  Beams  to  resist  Flexure. 

The  stiffness  of  beams  of  the  same  length  and 
width  varies  as  the  cube  of  the  depth.  If  the  depth 
is  the  same,   the  stiffness  varies  as  the  width. 

60.  There  are  two  beams,  of  equal  length  and 
width.  One  is  8  inches  in  depth;  the  other,  11 
inches.  If  it  require  30  cwt.  to  bend  the  former  1 
inch,  what  weight  will  it  require  to  bend  the  latter  1 
inch  ? 

61.  There  is  a  stick  of  timber  8  inches  by  6.  If  it 
require  24  cwt.  to  bend  it  2  inches  when  lying  flat, 
what  weight  will  bend  it  2  inches  when  turned  up  on 
the  edge? 

62.  If  10  cwt.  will  bend  the  stick  just  described  l£ 
inches  when  it  lies  flat,  what  weight  will  be  requisite 
to  bend  it  1£  inches  when  turned  up  on  the  edge  ? 

63.  There  is  a  board  12  inches  wide,  and  1  inch  in 
thickness.  What  is  the  ratio  of  its  strength  when 
lying  flat,  supported  at  the  ends,  to  its  strength  when 
turned  edgewise  ? 

64.  If  it  require  12  lbs.  to  bend  the  same  board  £  an 
inch,  when  lying  flat,  how  much  will  it  require  to 
bend  it  £  an  inch  when  turned  edgewise  ? 

23 


266  BUSINESS    FORMS    AND    INSTRUMENTS. 

SECTION    XLV. 

BUSINESS   FORMS   AND   INSTRUMENTS. 
Promissory  Notes. 

1. — On  Demand,  with  Interest. 

$500.  —  Boston,  March  1,  1846.  For  value  re- 
ceived, I  promise  A.  B.  to  pay  him,  or  his  order,  five 
hundred  dollars,  on  demand,  with  interest.       T.  M. 

2. — On   Time,  with  Interest. 

$200.  —  Boston,  March  1,  1846.  For  value  re- 
ceived, I  promise  A.  B.  to  pay  him,  or  his  order,  two 
hundred  dollars,  in  three  months,  with  interest. 

T.  M. 
3. — On   Time,  without  Interest. 

$400. — Boston,  March  1,  1846.  For  value  re- 
ceived, I  promise  A.  B.  to  pay  him,  or  his  order,  four 
hundred  dollars,  in  sixty  days  from  date.  I.  M. 

4.  — Payable  by  Instalments,  with  Periodical  Interest 

$1000.  — Boston,  March  1,  1846.  For  value  re- 
ceived, I  promise  A.  B.  to  pay  him,  or  his  order,  one 
thousand  dollars,  as  follows,  viz. ;  —  two  hundred  dol- 
lars in  one  year,  two  hundred  dollars  in  two  years,  and 
six  hundred  dollars  in  three  years,  from  this  date,  with 
interest  semi-annually.  I.  M. 

Remarks  on  Promissory  Notes. 

When  the  words  "or  order"  are  inserted  in  a  note, 
the  holder  of  the  note  may  endorse  it,  that  is,  write 


BUSINESS    FORMS    AND    INSTRUMENTS.  267 

his  name  on  the  back  of  it,  and  pass  it  to  a  third  per- 
son, who  can  collect  it  in  the  same  manner  as  if  he 
were  the  original  holder.  If  the  maker  of  the  note 
neglects  to  pay,  the  holder  may  collect  it  of  the  en- 
dorser. 

If  the  words  "or  bearer"  are  inserted  instead  of 
"or  order,"  any  person  who  has  possession  of  the  note 
may  collect  it  of  the  maker.  Such  a  note  would  be 
like  a  bank  note,  which  passes  from  hand  to  hand 
without  endorsement. 

A  note,  in  order  to  be  legal  in  the  first  holder's 
hands,  must  be  for  value  received.  A  note,  therefore, 
given  to  pay  a  debt  incurred  in  gambling  or  betting, 
cannot  be  collected  by  law,  unless  it  has  passed  into 
the  hands  of  an  innocent  holder. 

When  a  note  contains  the  promise  to  pay  interest 
annually,  and  the  interest  is  not  collected  annually,  the 
law  does  not  permit  the  holder  to  draw  compound  in- 
terest. The  holder  may  compel  the  payment  of  the 
interest  when  it  becomes  due ;  but  if  he  neglect  to  do 
this,  he  can  recover  only  simple  interest. 

When  a  note  is  given  to  pay  in  a  certain  commodity, 
as  wood,  grain,  &c,  if  the  note  is  not  paid  when  due, 
the  holder  may  compel  the  payment  of  the  equitable 
value  of  the  commodity  in  money.  The  reason  of 
this  is,  that  it  is  supposed  that  the  commodity  may 
have  a  value  to  the  holder  at  the  time  when  it  is 
promised,  which  it  will  lose  if  not  paid  then. 


Receipts. 

1.  —  A  general  Form. 

$500.  — Boston,  March  1,  1846.  Received  of 
O.  P.  the  sum  of  five  hundred  dollars,  in  full  of  all 
demands  against  him.  A.  B. 


268  BUSINESS    FORMS    AND    INSTRUMENTS. 

2.  —  For  Money  paid  by  another  Person. 

$300. —  Boston,  March  1,  1846.  Received  of 
O.  P.,  by  the  hand  of  Y.  Z.,  three  hundred  dollars,  in 
full  payment  for  a  chaise  by  me  sold  and  delivered  to 
the  said  O.  P.  A.  B. 

3.  —  For  Money  received  for  Another. 

$700. —Boston,  March  1,  1846.  Received  of 
O.  P.  seven  hundred  dollars,  it  being  for  the  balance 
of  account  due  from  said  O.  P.  to  Y.  Z.  A.  B. 

4.  —  In  Part  of  a  Bond. 

$3000. —  Boston,  March  1,  1846.  Received  of 
O.  P.  the  sum  of  three  thousand  dollars,  being  a  part 
of  the  sum  of  five  thousand  dollars  due  from  said  O.  P. 
to  me  on  the day  of .  A.  B. 

5.  —  For  Interest  due  on  a  Pond. 

$600.  —Boston,  March  1,  1846.  Received  of 
O.  P.  six  hundred  dollars,  due  this,  day  from  him  to 
me,  as  the  annual  interest  on  a  bond,  given  by  said  O. 
P.  to  me  on  the  1st  of  May,  1831,  for  the  payment  to 
me  of  ten  thousand  dollars  in  three  years,  with  inter- 
est annually.  A.  B. 

6.  —  On  Account. 

$50.  —  Boston,  March  1, 1846.  Received  of  O.  P. 
fifty  dollars,  for  which  I  promise  to  account  to  him  on 
a  settlement  between  us.  A.  B. 

7.  —  Of  Papers. 

Boston,  March  1,  1846.  Received  of  O.  P.  several 
contracts  and  papers,  which  are  described  as  follows:  — 
[describe  the  papers  ;]  which  I  promise  to  return  to  the 
said  O.  P.  on  demand.  A.  B. 


business  forms  and  instruments.  269 

Order  at  Sight. 

Boston,  April  18,  1846.  At  sight,  pay  to  the  order 
of  John  Brown,  one  thousand  dollars,  value  received, 
which  place  to  account  of 

Your  obedienf  servants,  A.  W.  &  Co. 

Jacob  Smith,  Esq.,  New  York. 

Order  on  Time. 

Boston,  April  18,  1846.  Six  months  after  date,  pay 
to  the  order  of  John  Brown,  one  thousand  dollars,  value 
received,  which  place  to  the  account  of 

Your  obedient  servants,  A.  W.  &  Co. 

Jacob  Smith,  Esq.,  New  York. 

Remarks.  —  If  J.  B.  present  this  order  to  J.  S.,  and  J.  S.  write  his 
name  across  the  face  of  it,  it  becomes  what  is  called  an  acceptance. 
J.  S.  agrees  to  pay  it  at  the  date  named. 

If  J.  B.  writes  his  name  on  the  back  of  the  acceptance,  it  becomes 
negotiable.  He  may  pass  it  to  a  third  person,  who  may  endorse  it, 
and  pass  it  to  a  fourth.  All  those  whose  signatures  are  on  the  order 
are  bound  for  its  payment ;  —  the  acceptor  to  the  drawer ;  the  acceptor 
and  drawer  to  the  first  endorser  ;  and  they  and  each  endorser  to  the 
one  succeeding  him ;  and  the  last  endorser,  and  all  previous  parties, 
to  the  holder. 


Award  by  Referees. 

We,  the  undersigned,  appointed  by  agreement  of  the 
parties  herein  named,  having  met  the  parties,  and  heard 
their  several  allegations,  arguments,  and  proofs,  and 
duly  considered  the  same,  do  award  and  determine  that 
A.  B.  shall  recover  of  C.  D.  the  sum  of ,  to- 
gether with  all  the  costs  of  this  reference,  which  are 

to  the  amount  of ;  and  that,  this  shall  be  final 

and  in  full  of  all  claims  and  dues  of  the  parties  on 
matters  herein  referred  to  us.  I.  M. 


R.  N. 
L.  S. 


23 


270  business  forms  and  instruments. 

Letter  of  Credit  for  Goods. 

Boston,  March  1,  1846. 
Messrs.  Y.  &  Z.,  Merchants,  Baltimore. 

Gentlemen,  —  Please  to  deliver  Mr.  C.  D., 

of ,  or  to  his  order,  gt>ods  and  merchandise  to  an 

amount  not  exceeding  in  value,  in  the  whole,  one  hun- 
dred dollars ;  and,  on  your  so  doing,  I  hereby  hold  my- 
self accountable  to  you  for  the  payment  of  the  same, 
in  case  Mr.  C.  D.  should  not  be  able  so  to  do,  or  should 
make  default,  of  which  default  you  are  required  to  give 
me  reasonable  and  proper  notice. 

Your  obedient  servant,  A.  B. 

A  letter  of  credit  for  money  may  be  given  in  the  general  form  of 
the  above ;  specifying,  in  the  letter,  the  amount  of  credit  granted. 


Power  of  Attorney. 
Know  all  men  by  these  presents,  that  I,  A.  B.,  of 
-,  do  hereby  appoint  C.  D.,  of ,  to  be  my 


sufficient  and  lawful  attorney,  to  act  for  me,  and  in  my 
name,  [here  slate  the  ohjects  for  which  he  is  to  act.]  And 
for  the  purposes  aforesaid,  I  hereby  grant  unto  my  said 
attorney  full  power  to  execute  all  needed  legal  instru- 
ments, to  institute  and  prosecute  all  claims  in  my  be- 
half, to  defend  all  suits  against  me,  to  submit  to  arbi- 
tration, or  settle  all  matters  in  dispute,  and  to  do  all 
such  acts  as  he  shall  think  expedient  for  the  full  ac- 
complishment of  the  objects  for  which  he  is  appointed 
my  attorney,  as  fully  as  I  might  myself  do  them  if 
present ;  and  all  acts  done  by  the  said  C.  D.,  my  attor- 
ney, under  authority  of  this  appointment,  I  will  ratify 
and  confirm. 

In  testimony  whereof,  I  hereby  set  my  hand  and 

seal,  this day  of ,  in  the  year . 

A.  B.     [l.  s.] 

Signed,  sealed,  and  delivered, 
in  the  presence  of   S.  N. 
W.  F. 


THE    STANDARD    OF    WEIGHTS    AND    MEASURES.     271 

SECTION    XLVI. 

ON  THE  STANDARD   OF  WEIGHTS   AND   MEASURES. 

In  the  earlier  states  of  society,  the  standard  of 
weights  and  measures  was,  of  necessity,  very  indefi- 
nite and  fluctuating.  In  one  nation  it  was  one  thing; 
in  another  nation,  another  j  and  in  no  case  was  it  de- 
serving of  a  very  high  degree  of  confidence. 

Sometimes  the  length  of  the  king's  foot  was  the 
standard  for  all  measures  of  length ;  again,  the  length 
of  the  king's  arm,  from  the  elbow  to  the  extremity  of 
the  fingers,  was  made  the  standard. 

The  length  of  journeys  was  measured  by  the  hours 
or  days  employed  in  performing  them,  or  by  the  num- 
ber of  steps  taken. 

In  land  measure,  the  standard  was,  what  a  yoke  of 
oxen  could  plough  in  a  day,  when,  in  fact,  one  yoke 
might  plough  twice  as  much  as  another. 

In  dry  measure,  it  was  as  much  as  a  man  could 
conveniently  carry,  without  first  deciding  how  strong 
the  man  should  be. 

In  weight,  the  standard  was,  what  a  man  could  hold 
and  swing  in  his  hand. 

Sometimes  vegetables  were  taken  as  measures;  as, 
"three  barley  corns  make  one  inch."  But  barley 
corns  do  not  all  grow  of  exactly  equal  length,  any 
more  than  the  feet  and  arms  of  kings. 
.  As  science  advanced,  and  commerce  became  far- 
Uher  extended  among  different  nations,  the  mischiefs 
of  these  vague  and  fluctuating  methods  of  measure- 
ment became  more  and  more  deeply  felt. 

But  it  was  far  easier  to  see  the  faults  of  the  old  sys- 
tem, than  to  devise  a  new  one  that  should  be  perfect. 
What  object  could  be  selected  as  an  ultimate  standard 
for  all  weights  and  measures? 


272      THE    STANDARD    OF    WEIGHTS    AND    MEASURES. 

We  have  seen  that  the  parts  of  animals  or  of  vege- 
tables are  too  liable  to  change  to  deserve  any  confi- 
dence. If  some  arbitrary  standard  should  be  adopted, 
as  a  foot,  or  yard,  and  this  measure  should  be  kept  as 
the  standard,  by  which  all  others  should  be  tried,  what 
security  could  there  be  that  it  would  never  be  altered 
by  fraud  or  destroyed  by  accident  ?  Or,  if  some  natu- 
ral distance  were  taken,  as  the  distance  between  two 
points  of  some  well-known  rock  or  cliff,  this  distance 
might  vary  with  a  change  of  temperature  or  be  altered 
by  some  convulsion  of  nature. 

We  will  proceed  to  give  a  short  account  of  the 
English  system  of  weights  and  measures,  adopted  by 
their  Act  of  Uniformity,  which  took  effect  Jan.  1, 1826. 
To  begin  with  measures  of  capacity ;  all  English 
measures  of  capacity,  whether  for  liquors  or  grain, 
are  referred  to  the  standard  imperial  gallon.  This 
gallon  contains  277i  cubic  inches.  From  this  gallon, 
quarts,  pints,  and  gills,  are  obtained  by  subdivision  ; 
and  pecks  and  bushels  by  multiplication.  Hence  you 
can  find  the  number  of  cubic  inches  in  an  English 
quart,  pint,  peck,  or  bushel.  Thus  the  adoption  of  the 
imperial  gallon  introduces  entire  uniformity  into  all 
English  measures  of  capacity.  It  refers  them  all 
ultimately  to  the  cubic  inch.  We  must  now  inquire, 
what  has  been  done  to  fix  the  measure  of  the  inch ; 
for,  if  there  is  any  error  or  variation  here,  it  will  ren- 
der false  all  the  measures  of  capacity  which  depend 
upon  it. 

To  determine  the  measure  of  the  inch,  it  is  made 
by  law  -gV  of  the  standard  yard.  That  standard  yard* 
is  a  straight  brass  rod  in  the  custody  of  the  clerk  of 
the  House  of  Commons.  The  yard  is  the  distance 
on  that  rod  between  the  centres  of  the  points  in  the 
two  gold  studs  or  pins  in  the  rod.  And  as  heat  would 
make  the  yard  longer,  and  cold  would  make  it  shorter, 
the  law  requires  that  it  shall  be  used  wrhen  it  is  of  the 


THE    STANDARD    OF    WEIGHTS    AND    MEASURES.      273 

temperature  of  62°  (Fahrenheit.)  This  standard  yard, 
however,  may  be  destroyed  by  accident.  We  must 
then  inquire  for  a  still  more  permanent  standard.  To 
effect  this,  the  law  declares  that  the  standard  yard,  if 
destroyed,  may  be  restored,  by  making  it  ttt8M  °f  tne 
length  of  a  pendulum,  that  vibrates  seconds  in  the  lati- 
tude of  London,  in  a  vacuum,  at  the  level  of  the  sea.: 
If  all  these  conditions  are  fulfilled,  a  pendulum  that 
vibrates  seconds  must  have  an  absolutely  invariable 
length. 

Thus  we  have  brought  the  whole  system  of  meas- 
ures back  to  seconds,  as  the  standard.  The  whole 
scheme  now  depends  upon  seconds  being  of  an  in- 
variable length. 

Seconds  are  parts  of  a  year.  The  year  is  not  made 
up  by  multiplying  seconds,  but  seconds  are  obtained 
by  dividing  the  year.  If,  then,  the  year  is  of  a  fixed 
length,  seconds  are  so.  Now,  the  year  is  the  time  of 
the  revolution  of  the  earth  round  the  sun.  It  is  the 
same,  without  change,  from  one  year  to  another,  and 
from  century  to  century. 

Thus  the  whole  system  of  measures  has  been 
brought,  for  its  ultimate  standard,  to  the  unalterable 
period  of  the  earth's  revolution  round  the  sun. 

We  will  now  retrace  the  steps  of  this  investiga- 
tion, beginning  with  the  primary  standard,  the  earth's 
yearly  revolution. 

The  time  of  the  earth's  revolution  round  the  sun  is 
always  the  same.  Therefore,  a  second,  which  is  a 
certain  part  of  this  time,  is  an  exact  measure.  If  the 
second  is  a  fixed  measure,  then  the  pendulum  which, 
under  the  same  circumstances,  vibrates  seconds,  is  of 
a  fixed  length.  If  the  length  of  the  pendulum  vibra- 
ting seconds  is  fixed,  the  length  of  the  standard  yard 
is  fixed,  for  it  is  $f$$$$  of  the  pendulum.  If  the 
standard  yard  is  fixed,  the  inch  is  fixed  ;  consequently 
the  cubic  inch,  the  gallon,  quart,  pint,  gill,  and  bushel. 

s 


274  THE  STANDARD  OF  WEIGHTS  AND  MEASURES. 

In  the  preceding  investigation,  no  mention  has  been 
made  of  the  standard  of  weight.  It  is  obtained  by 
making  a  cubic  inch  of  distilled  water  equal  to 
252.458  grs.,  of  which  5760  make  a  pound  Troy, 
and  7000  make  a  pound  avoirdupois. 

Thus  weights  and  measures  are  alike  brought  to  an 
unalterable  standard. 

The  imperial  gallon  contains 277^  cubic  inches. 

The  Winchester*  gal.,  wine  measure,  231       "  " 

"  "  "    beer  measure,  282       "  " 

The  imperial  gallon  of  water  weighs  1 0  /6s.  avoirdupois. 

The  system  of  weights  and  measures  established 
by  law  in  the  United  States,  is  very  nearly  the  same 
as  the  English.  The  gallon,  United  States  measure, 
contains  9  lbs.  14  oz.  of  water.  This  is  the  legal 
standard  for  all  measures,  dry  and  liquid.  In  many 
parts  of  the  country,  however,  especially  in  the  interior, 
the  legal  standard  has  not  supplanted  the  system  de- 
rived in  earlier  times  from  the  English. 

In  France,  where  th§  system  of  weights  has  been 
carried  to  greater  perfection  than  in  any  other  country, 
the  decimal  ratio  is*  adopted  in  all  denominations.  In 
some  cases,  however,  there  is  still  retained  some  part 
of  the  old  system,  combined,  with  the  decimal. 

In  obtaining  an  ultimate  standard  of  measure,  the 
French  measured  one  quarter  of  a  meridian  line  of 
longitude. 

One  ten  millionth  part  of  this  arc  they  made  the 
basis  of  their  system  of  measures.  This  standard, 
the  metre,  is  3.28  feet.  The  lower  denominations  are 
made  by  successive  divisions  of  this,  by  10,  100,  &c, 
and  the  -higher  by  multiplication. 


*  Winchester ;   so  called  because  the  standard  measures  were  kept 
*at  Winchester. 


THE    STANDARD    OF    WEIGHTS    AND    MEASURES.      275 

The   following   table  presents  the  French  decimal 
weights  and  measures,  with  the  English  equivalents. 

French  Long  Measure. 

feet. 

10  millimetres  make  ....  1  centimetre, 0328 

10  centimetres, 1  decimetre, 328 

10  decimetres, 1  metre, 3.28 

10  metres, 1  decametre, 32.8 

10  decametres, 1  hectometre, 328 

10  hectometres, 1  kilometre, 3280 

10  kilometres, 1  myreametre,  ....  32800 


French  Square  Measure. 

The  unit  square  measure  is  the  are,  which  is  the 
square  of  the  decametre  ;  consequently,  it  is  the  square 
of  32.8  feet,  —  a  little  less  than  4  square  rods. 

This  unit  is  multiplied  for  the  higher  denomina- 
tions, and  divided  for  'the  lower,  in  the  same  way 
as  the  metre. 


French  Decimal  Weight. 


10  milligrammes  make  .  1 
10  centigrammes, ......  1 

10  decigrammes, 1 

10  grammes, 1 


10  decagrammes, 1 

10  hectogrammes,  .  .  .  .  1 

10  kilogrammes, 1 

10  myriagrammes, ....  1 
10  quintals, 1 


decigramme, .  , 
gramme,    .  .  . 
decagramme,    , 
hectogramme, 
kilogramme,  .  , 
myriagramme, 
quintal,    .... 
million , 


grs.  Trov. 

.1543402 
1.543402 
15.43402 
154.3402 
1543.402 
15434.02 
154340.2 
1543402. 
15434020. 


276  MISCELLANEOUS    EXAMPLES. 


MISCELLANEOUS    EXAMPLES. 

The  examples  that  follow  are  designed  to  carry 
still  farther  the  practice  in    Written  Arithmetic. 

1.  James  Ball  bought  of  Amos  Sewall  three  pieces 
broadcloth,  measuring  12^,  13,  and  24  yards,  at 
$4.87^  per  yard;  five  pieces  kerseymere,  measuring 
24£,  25,  27,  26i,  and  26  yards,  at  67  cts.  per  yard ; 
eight  pieces  cotton  sheeting,  measuring  33  yards  each, 
at  9£  cents  per  yard ;  £  per  cent.  off.  What  was  the 
amount  of  the  bill  ? 

2.  Bought  of  John  Jones,  on  six  months'  credit,  47 
yards  broadcloth,  at  $4.31  per  yard;  16  yards  vest- 
ings,  at  $1.15  per  yard  ;  63^  yards  satinet,  at  62^  cents 
per  yard ;  5J-  pieces  sheeting,  containing  33  yards  a 
piece,  at  8f  cents  per  yard.  John  Jones  agrees,  if  paid 
in  cash,  to  deduct  4  per  cent,  from  the  bill.  What  is 
the  cash  amount  of  the  bill  ? 

3.  Bought  of  Asa  Wood,  on  six  months'  credit,  45 
barrels  flour,  at  $5.37  per  barrel;  four  hhds.  molasses, 
containing  124^  131,  134,  and  136  gallons,  at  27£ 
cents  per  gallon ;  five  bags  coffee,  containing  54£,  56, 
49£,  62,  and  65^-  lbs.,  at  8£  cents  per  pound.  Gave,  in 
payment  of  the  above,  a  note  payable  in  six  months. 
What  was  the  cash  value  of  the  note  when  it  was 
given,  reckoning  interest  at  6  per  cent.  ? 

4.  A  bought  of  B,  on  six  months'  credit,  goods  with 
the  amount  as  follows  :  — 

April  3,  1845, $254.75. 

June  8,       "      135.00. 

Aug.  1,      "      v  .  .  .  .  200.00. 

♦Sept.  14,   "      168.25. 

At  what  date  shall  A  make  an  equated  payment  of  the 
whole  amount  ? 


MISCELLANEOUS     EXAMPLES.  277 

5.  A  gives  B  a  note  for  $600,  payable  in  six  months. 
What  is  the  cash  value  of  the  note,  two  months  after 
date,  reckoning  the  interest  at  6  per  cent.  ? 

6.  A  man  agrees  to  dig  and  stone  a  well  on  the  fol- 
lowing terms: — $1.00  per  foot  for  the  first  10  feet, 
$2.50  per  foot  for  the  second  10  feet,  and  $4.00  per 
foot  for  the  remainder,  till  he  finds  a  supply  of  water. 
For  every  foot  of  rock  through  which  he  digs,  he 
shall  receive  double  pay.  He  digs  42  feet  in  all,  and 
through  rock  from  17  to  31£  feet  from  the  surface. 
What  pay  is  he  entitled  to  for  the  whole  ? 

7.  A  man  engages  to  build  160  rods  of  road,  one  half 
for  $1.42  per  rod,  the  other  half  for  $1.83  per  rod. 
He  hires  93  days  of  men's  labor  at  84  cents  per  day ; 
pays  for  board  of  the  same  at  $1.50  per  week  of  six 
days;  pays  for  tools  and  repairs,  $11.60.  He  works 
himself,  with  4  oxen  and  his  son,  34  days.  What 
wages  will  he  receive  per  day  for  himself,  for  his  oxen, 
and  for  his  son,  allowing  for  the  4  oxen  as  much  as  for 
himself,  and  for  his  son  half  as  much  ? 

8.  A  bought  a  lot  of  standing  wood  for  105  dollars, 
and  agreed  with  B  to  cut  and  haul  it  to  market  for 
three  fifths  of  the  proceeds.  There  were  54  cords  of 
pine,  which  was  sold  for  $2.84  per  cord,  and  61  cords 
of  hard  wood,  which  sold  for  $4.75  per  cord.  Did  A 
gain,  or  lose,  and  how  much?  B  labored  himself  35 
days,  employed  one  yoke  of  oxen  24  days,  and  hired 
68  days  of  men's  labor  at  85  cents  per  day.  What 
pay  does  he  receive  per  day  for  himself  and  for  his 
oxen,  allowing  for  his  oxen,  per  day,  two  thirds  as 
much  as  for  himself? 

9.  A  and  B  bought  a  quantity  of  grass,  ready  for 
cutting,  for  26  dollars,  for  which  they  paid  13  dollars 
apiece.  In  cutting  and  curing  it,  A  furnished  3  days 
of  hired  men's  labor,  and  worked  himself  2J  days;  B 
hired  7  days  of  men's  labor,  a  team  1J-  days,  and 
worked  himself  24-  days.     There  were  6 \  tons  of  hay. 

24 


278  MISCELLANEOUS    EXAMPLES. 

What  is  each  one's  share  of  the  hay,  allowing  for  the 
labor  $1.00  per  clay,  and  for  the  whole  work  of  the 
team  $1.50? 

10.  A  agrees  to  dig  and  stone  a  cellar  7  feet  deep. 
It  is  to  be  13  feet  wide  and  16  feet  long  inside  the 
walls,  which  are  to  be  2  feet  in  thickness.  He  is  to 
receive  for  digging  22  cents  a  cubic  yard  for  earth,  and 
$1.55  a  cnbic  yard  for  rock,  and  for  stoning  S7  cents 
for  every  perch  of  25  cubic  feet  for  which  the  stone  is 
found  in  digging  the  cellar,  and  $1.50  a  perch  when 
he  has  to  bring  the  stone  from  another  place.  In  dig- 
ging the  cellar,  he  digs  7^  cubic  yards  of  rock,  which 
furnishes  stone  for  7£  cubic  yards  of  wall.  What  is 
he  to  receive  for  the  whole  job  ? 

11.  A  man  agrees  to  dig  a  canal  10  rods  long,  6  feet 
in  depth,  33  feet  wide  at  the  top,  and  24  feet  wide  at 
the  bottom,  for  10  cents  per  cubic  yard.  What  sum 
will  the  work  amount  to  ? 

12.  What  is  the  value  of  six  loads  of  wood,  at 
$4.67  per  cord,  measuring  as  follows:  —  first,  8  ft.,  4 
ft.  3  in.,  3  ft.  10  in. ;  second,  8  ft.  4  in.,  4  ft.  1  in., 
4  ft. ;  third,  8  ft.  2  in.,  4  ft.  7  in.,  3  ft.  11  in. ;  fourth, 
8  ft.,  4  ft.  2  in.,  4  ft. ;  fifth,  8  ft.  2  in.,  3  ft.  10  in., 
3  ft.  9  in.  ;  sixth,  8  ft.  6  in.,  4  ft.,  4  ft.  4  in.  ? 

13.  What  will  be  the  dimension,  at  the  two  ends,  of 
the  largest  square  stick  of  timber  that  can  be  hewn 
from  a  round  log  3  feet  in  diameter  at  the  larger  end, 
and  2  feet  in  diameter  at  the  smaller? 

14.  What  will  be  the  solid  contents  of  such  a  stick, 
if  it  is  16  feet  in  length  ? 

15.  What  will  be  the  width  at  the  two  ends  of  the 
largest  stick  of  timber  that  can  be  hewn  from  a  log 
3  feet  in  diameter  at  the  larger  end,  and  2  feet  in 
diameter  at  the  smaller,  if  the  stick  is  hewn  10  inches 
in  thickness  through  its  whole  length  ? 

16.  What  will  be  the  solid  contents,  of  such  a  stick, 
if  it  is  20  feet  in  length  ? 


MISCELLANEOUS    EXAMPLES.  279 

17.  There  are  three  pieces  of  cloth.  The  length  of 
the  first  is  to  that  of  the  second  as  3  to  2,  the  length 
of  the  second  is  to  that  of  the  third  as  4  to  15,  and 
the  length  of  the  three  added  together  is  50  yards. 
What  is  the  length  of  each  piece  ? 

18.  A  man  bought  a  chaise,  and  paid  20  dollars  for 
repairing  it.  He  then  sold  it  for  one  fifth  more  than 
he  gave,  and  found  that,  allowing  one  dollar  for  his 
own  trouble  in  the  business,  he  had  lost  thirteen  dol- 
lars.     What  did  he  give  for  the  chaise  ? 

19.  A  gentleman  began  his  preparation  for  college 
at  a  certain  age,  and  spent  in  school  and  at  college 
half  as  many  years  as  he  had  lived  before.  He  then 
went  to  Europe,  and,  after  spending  there  one  ninth 
as  many  years  as  his  age  amounted  to  when  he  left 
Europe,  spent  in  his  profession  one  third  as  many 
years  as  he  had  lived  when  he  entered  it,  and  was  then 
36  years  old.  At  what  age  did  he  begin  his  prepara- 
tion for  college  ? 

20.  One  half  of  three  fourths  of  A's  age  equals  one 
sixth  of  B's  age,  and  the  sum  of  their  ages  is  78. 
What  is  the  age  of  each  ? 

21.  Three  fourths  of  the  liquor  in  a  cask  equals  five 
sixths  of  what  has  leaked  out ;  and  the  whole,  before 
any  leaked  out,  was  sixty  gallons.  How  much  is 
there  in  the  cask? 

22.  Reduce  5  pence  to  the  decimal  of  a  shilling, 
carrying  the  decimal  to  the  sixth  figure. 

23.  Reduce  9£  gallons  to  the  decimal  of  a  barrel, 
wine  measure,  carrying  the  decimal  to  the  tenth 
figure. 

24.  Reduce  7%  quarts  to  the  decimal  of  a  bushel. 

25.  What  is  the  least  common  multiple  of  784, 
1386,  and  1235? 

26.  What  are  the  prime  numbers  between  1010  and 
1020? 

27.  What  are  the  prime  factors  of  2326? 


280  MISCELLANEOUS    EXAMPLES. 

1  T      1  "  1  91 

28.  Reduce  — — ,  — —  f  and  — — ,  to  simple  fractions, 

6&-$    19  30^- 

with  a  common  denominator. 

29.  What  is  the  value,  in  shillings  and  pence,  of 
the  following  decimals,  when  added  together :  — 
£.0431;  .67142s.;  .73462d.? 

30.  What  is  the  interest  of  $642.25  for  1  year  and 
3  months,  at  7i  per  cent.  ? 

31.  What  is  the  interest  of  $954.30  for  2  years,  4 
months,  and  21  days,  at  5  per  cent.  ? 

32.  What  is  the  present  worth  of  a  note  of  $640.50 
due  in  3  months  ? 

33.  What  is  the  present  worth  of  a  note  of  $  1263.00 
due  in  3£  months,  at  5  per  cent,  interest  ? 

34.  Boston,  June  14,  1836.  For  value  received,  I 
promise  to  pay  John  Ball,  or  order,  three  hundred  and 
sixty-five  dollars  in  four  years  with  interest  annually. 

James  Frost. 
If  no  interest  is  paid,  and  the  note  is  renewed  an- 
nually, what  will  be  the  amount  of  the  note  four  years 
after  date  ? 

35.  New  York,  July  1,  1840.  For  value  received, 
I  promise  to  pay  Abel  Jones,  or  order,  five  hundred  and 
forty  dollars,  on  demand,  with  interest. 

John  Frost. 

Endorsements,  —  Oct.  3,  1840, $63.00 

Feb.  4,  1841, 120.00 

June  1,  1841, 60.00 

Sept.  15,  1841, 200.00 

What  will  be  due  Feb.  14,  1842,  at  seven  per  cent. 
interest  ? 

36.  A  owes  B  $400.00,  due  in  2  months;  $320.00, 
due  in  3  months;  $600.00,  due  in  4£  months.  What 
is  the  equated  time  for  the  payment  of  the  whole  ? 

37.  What  is  the  present  worth  of  a  bank  note  for 
800  dollars  payable  in  three  months  ? 


MISCELLANEOUS    EXAMPLES.  281 

38.  What  is  the  present  worth  of  a  bank  note  for 
346  dollars  payable  in  six  months? 

39.  For  what  sum  must  I  give  a  note  to  a  bank 
payable  in  three  months,  in  order  to  obtain  674  dol- 
lars? 

40.  Bought  seven  100  dollar  shares  of  bank  stock 
at  6£  per  cent,  advance.  Gave  in  payment  nine  60 
dollar  shares  of  railroad  stock  at  3  per  cent,  discount, 
and  a  bank  note  payable  in  60  days.  What  was  the 
face  of  the  note  ? 

41.  How  many  square  inches  are  there  in  15^-  square 
rods  ? 

42.  What  is  the  cost  of  plastering  the  sides,  ends, 
and  ceiling  of  a  room  22J-  feet  long,  17f  feet  wide,  and 
11  feet  1  inch  in  height,  at  18  cents  per  square  yard, 
making  no  deduction  for  windows,  doors,  or  wood- 
work ? 

43.  There  is  a  house  40  feet  in  length,  and  26  feet 
in  breadth.  From  the  beam  to  the  ridgepole  is  11  feet. 
The  roof  projects  7  inches  beyond  the  walls  at  each 
end,  and  the  line  of  the  eaves  is  8  inches,  measured 
horizontally,  from  the  side  walls.  How  many  square 
feet  are  there  in  the  roof? 

44.  A  painter  agrees  to  paint  the  outside  of  a  house 
which  is  44  feet  long,  28  feet  wide,  22£  feet  in  height 
to  the  top  of  the  beam,  and  11  feet  8  inches  from  the 
beam  to  the  ridgepole,  for  44  cents  per  square  yard. 
What  is  he  entitled  to  for  the  job,  making  no  deduc- 
tion for  windows,  and  no  addition  for  cornices  or  other 
projections  ? 

45.  What  is  the  square  root  of  9743  to  three  places 
of  decimals? 

46.  What  is  the  square  root  of  17431  to  two  places 
of  decimals? 

47.  The  base  of  a  right-angled  triangle  is  744  feet, 
the  hypotenuse  834  feet.  What  is  the  perpendicular, 
to  two  places  of  decimals? 

24* 


282  MISCELLANEOUS    EXAMPLES. 

48.  The  base  of  a  triangle  is  76  rods,  the  sum  of 
the  hypotenuse  and  perpendicular  is  186  rods.  What 
is  the  length  of  the  hypotenuse  ? 

49.  From  a  cylinder  12  inches  in. diameter,  it  is  de- 
sired to  cut  the  largest  possible  four-sided  prism,  whose 
opposite  sides  shall  be  parallel,  and  whose  width  shall 
be  to  its  thickness  as  2  to  1.  What  will  be  its  width, 
and  what  its  thickness? 

50.  What  are  the  dimensions  of  the  largest  prism, 
with  parallel  sides,  that  can  be  cut  from  a  cylinder  12 
inches  in  diameter,  making  the  width  to  the  thickness 
as  3  to  1  ?  * 

51.  What  are  the  dimensions  of  the  largest  prism, 
with  parallel  sides,  that  can  be  cut  from  a  sphere  15 
inches  in  diameter,  making  the  length  and  breadth 
equal,  and  each  of  them  double  of  the  thickness  ? 

52.  What  is  the  cube  root  of  674  to  three  places 
of  decimals? 

53.  What  is  the  cube  root  of  1736  to  two  places 
of  decimals  ? 

54.  What  is  the  cube  root  of  31  to  three  places  of 
decimals  ? 

55.  Two  men  purchase  a  lot  of  land  for  750  dollars. 
One  pays  $406.50;  the  other,  the  remainder.  They 
expend  $341  in  equal  shares  on  its  improvement,  and 
sell  the  land  for  $1430.00.  What  is  each  one's  share 
of  the  gain? 

56.  1841  are  how  many  times  four-fifths  of  76£  ? 

57.  How  many  bottles,  each  containing  1$  pints, 
can  be  filled  from  a  hogshead  containing  63  gallons, 
allowing  a  loss  of  one  eleventh  in  the  process  ? 

58.  What  is  the  value,  in  Federal  money,  of  £456 
at  9J-  per  cent,  advance  ? 

59.  How  many  gallons,  each  containing  231  cubic 
inches,  will  fill  a  cylindrical  cistern  4  feet  in  diameter 
and  5  feet  deep? 

60.  There  is  a  cylindrical  cistern  6  feet  deep,  con- 


MISCELLANEOUS    EXAMPLES.  283 

taining  10  barrels  of  31  J-  gallons  each,  each  gallon 
containing  231  cubic  inches.  What  is  the  diameter 
of  the  cistern  ? 

61.  There  is  a  cistern,  in  the  form  of  an  inverted 
cone,  8  feet  deep,  and  of  the  same  capacity  as  the  cis- 
tern last  named.     What  is  its  diameter  at  the  top? 

62.  Bought  74  barrels  of  flour  at  $4.56  per  barrel, 
and,  after  keeping  it  35  days,  sold  it  at  $5.16  per  bar- 
rel. What  per  cent,  did  I  gain,  allowing  6  per  cent, 
interest  on  the  money  invested  ? 

63.  The  first  and  fourteenth  terms  of  an  arith- 
metical series  are  3  and  19.  What  is  the  common 
increase  ? 

64.  The  fifth  term  of  an  arithmetical  series  is  18, 
the  16th  term  is  39.     What  is  the  first  term? 

65.  What  is  the  sum  of  an  arithmetical  series,  the 
extremes  of  which  are  9  and  164,  and  the  number  of 
terms  forty? 

66.  Find  the  ninth  term  of  a  geometrical  series 
whose  first  term  is  2  and  ratio  f. 

67.  What  is  the  fifth  term  of  a  geometrical  series 
whose  second  term  is  4  and  ratio  f  ? 

6§.  James  Wildes  bought  of  John  Good, 

45£  bushels  Salt,  at  39£  cents  per  bushel; 
143|  lbs.  Rice,  at  3f  cts.  per  lb.  ; 

43J-  lbs.  Tea,  at  39  cts.  per  lb. ; 

94J  lbs.  Coffee,  at   101  cts.  per  lb. ; 

12     bbls.  Flour,  at   $5.87  per  bbl. 
What  is  the  amount  of  the  bill  ? 

69.  If  he  pays  the  above  bill  by  a  bank  note, 
discounted  for  60  days,  what  must  be  the  sum 
named  in  the  note  ? 

70.  Add  23f  +  18f+f  +  j|+| 

71.  Reduce  to  a  common  denominator,  23  and  £  of 
A  of  121  and  f  of  TV 

72.  Bought  3  boxes  of  sugar,  containing  3  cwt.  2 


284  MISCELLANEOUS    EXAMPLES. 

qrs.  17  lbs.;  3  cwt.  3  qrs.  11^  lbs.;  3  cwt.  2  qrs.  22} 
lbs. ;  at  6&  cts.  per  lb.  Paid  in  corn  at  57%  cts.  per 
bushel.     How  many  bushels  did  it  take? 

73.  What  is  the  solid  contents  of  a  wall  5  ft.  high, 
2  ft.  3  in.  wide  at  the  bottom,  and  1  foot  10  in.  wide 
at  the  top,  and  34  ft.  in  length? 

74.  Three  men  agreed  to  build  a  wall  6  ft.  high,  3 
ft.  wide  at  the  bottom,  and  2  ft.  wide  at  the  top. 
The  first  man  built  the  wall  to  the  height  of  2  feet 
from  the  ground,  the  second  raised  it  2  feet  more,  and 
the  third  finished  it.  What  proportional  share  of  the 
pay  ought  each  to  receive  ? 

75.  There  is  a  lever,  13  ft.  in  length,  which  is  sup- 
ported by  a  fulcrum  14  in.  from  the  end.  How  many 
pounds  applied  to  the  longer  end  will  balance  a  weight 
of  17  cwt.  2  qrs.  at  the  shorter  end  ? 

76.  The  axle  of  a  wheel  is  13  in.  in  diameter,  the 
wheel  is  11£  ft.  in  diameter.  What  power,  applied  to 
the  circumference,  will  balance  3^  tons  suspended  at 
the  axle? 

77.  If  the  threads  of  a  screw  are  l£  in.  apart,  what 
power,  applied  at  the  end  of  a  lever  9J-  ft.  in  length, 
will  support  7  tons,  allowing  nothing  for  friction? 

78.  With  the  same  screw,  what  power  would  sup- 
port 7  tons,  making  an  allowance  of  one  fourth  for 
friction?  Reflect  whether  the  friction  in  this  case  is 
in  favor  of  the  weight  or  in  favor  of  the  power. 

79.  With  the  same  screw,  what  power  would  raise 
7  tons,  allowing  one  fourth  for  friction?  Notice,  in 
this  case,  in  which  way  the  friction  will  operate. 

80.  There  are  two  right-angled  triangles  upon  a 
base  of  21  ft.  in  length  ;  the  perpendicular  of  the  larger 
is  9  ft.  in  length,  that  of  the  smaller  8£  ft.  What  is 
the  difference  in  the  length  of  the  hypotenuse  of  the 
two  triangles? 

81.  Divide  the  number  78  in  two  such  numbers 
that  the  first  shall  be  6  more  than  one  fifth  part  of  the 
second. 


MISCELLANEOUS     EXAMPLES.  285 

82.  Divide  the  number  82  into  two  such  parts  that 
the  first  diminished  by  5  shall  equal  one  sixth  part  of 
the  second. 

S3.  What  is  the  value,  in  Federal  money,  of  £194 
16s.,  at  8£  per  cent,  advance? 

84.  How  many  Winchester  gallons,  wine  measure, 
would  be  contained  in  a  cubical  vat  measuring  4 
ft.  each  way? 

85.  Divide  the  number  1520  into  three  such  parts 
that  twice  the  first  shall  be  40  less  than  the  second, 
and  the  second  shall  be  half  as  great  as  the  third. 

86.  How  many  yards  of  lining,  $  of  a  yard  wide, 
will  line  13£  yds.  of  cloth   lT7e  yds.  wide  ? 

87.  There  is  a  rectangular  field  containing  7£  acres, 
the  width  of  which  is  one  half  as  great  as  its  length. 
What  is  the  length  of  a  diagonal  line  connecting  its 
opposite  corners? 

88.  Around  a  rectangular  common,  containing  18 
acres,  the  length  of  which  is  to  its  breadth  as  6  to  5,  a 
road  runs  40  ft.  in  breadth.  How  many  rods  would 
be  saved  in  travel  by  crossing  the  common  diag- 
onally, rather  than  going  round  on  two  sides  of  it, 
supposing  the  traveller  to  begin  and  end,  in  both 
cases,  in  the  middle  of  the  road  in  range  with  the 
diagonal  line? 

89.  Whatsis  the  difference  between  the  square  root 
of  half  of  4,  and  half  the  square  root  of  4,  carried 
to  three  decimal  places? 

90.  What  is  tile  difference  between  the  square  root 
of  one  third  of  12,  and  one  third  the  square  root  of 
12,  carried  to  three  places  of  decimals? 

91.  How  many  times  £  of  17£  are  equal  to  13£ 
times  15f? 

92.  When  it  is  noon  in  Boston,  Lon.  71°  4'  W., 
what  time  is  it  at  Liverpool,  Lon.  2°  59'  W.;  at 
Greenwich,  Lon.  0;  at  Havre,  Lon.  0  16'  E.  ;  and 
at  Paris,  Lon.  2°  20'  E.  ?  ^ 


286  MISCELLANEOUS    EXAMPLES. 

93.  When  it  is  noon  at  London,  Lon.  0  W  W., 
what  time  is  it  at  New  York,  Lon.  74°  T  W. ;  at 
Washington,  Lon.  77°  2'  W.  ;  and  at  Cincinnati, 
Lon.  84°  27'  W.  ? 

94.  A  field,  in  the  form  of  an  equilateral  triangle, 
contains  8^  acres.  What  is  the  length  of  one  of 
its  sides? 

95.  Reduce  #  of  —  to  fifths.  Reduce  f  of  —  to 
fifteenths. 

96.  What  is  the  amount,  in  Federal  money,  of  6  s. 
7d.,  5s.  3d.,  14s.  9d.,  and  lis.  6}d.? 

97.  What  is  the  cube  root  of  144,  to  three  places  of 
decimals  ? 

98.  What  is  the  weight  of  a  cylinder  of  lead  3  ft. 
long  and  4  inches  in  diameter  ? 

13i  34A 

99.  Multiply  7J  times  ~  by  i  of  ~. 

100.  How  many  square  feet  in  a  triangle  whose 
base  measures" 45  feet,  and  whose  height  is  17  feet? 

101.  What  is  the  solid  contents  of  a  triangular 
pyramid  whose  base  measures  18  feet  on  each  side, 
and  whose  height  is  23  feet? 

102.  What  is  the  interest  of  $546.25  for  23  mo.  13 
days,  at  7£  per  cent.  ? 

103.  What  number  is  that  of  which  i  and  i  of  it 
added  together  exceed  £  of  it  by  2.? 

.  104.  What  is  the  cube  root  of  197  to  two  decimal 
places  ? 

105.  What  is  the  cube  root  of  501  to  four  decimal 
places  ? 

106.  What  is  the  43d  term  of  an  arithmetical  series 
whose  first  term  is  7|  and  common  difference  3f  ? 

107.  What  is  the  sum  of  an  arithmetical  series  of 
57  terms  whose  4th  term  is  15  and  common  dif- 
ference 2£? 

108.  How  many  pounds  of  coffee  at  11  cents  per 


MISCELLANEOUS    EXAMPLES.  287 

pound  can  be  mixed  with  56  lbs.  at  8  cents,  and  96 
lbs.  at  9J-  cents,  so  as  to  make  the  mixture  worth 
10i  cents? 

109.  If  a  sphere  of  gold  weigh  36  oz.,  how  many 
ounces  will  a  sphere  of  silver  weigh  of  equal  size,  the 
specific  gravity  being  as  given  on  page  258? 

110.  What  will  be  the  weight  of  a  ball  of  iron  6 
inches  in  diameter? 

111.  What  will  be  the  weight  of  the  largest  cube 
that  can  be  cut  from  a  ball  of  iron  6  inches  in  di- 
ameter? 

112.  What  is  the  value,  in  Federal  money,  of  £13 
6s.,  2  guineas,  3d.,  7^d.,  added  together? 

113.  How  many  times  will  a  wheel  4  ft.  3  in.  in 
diameter  go  round  in  traversing  the  circumference  of 
a  circle  containing  5  acres? 

114.  If  a  lever  is  16  ft.  in  length,  the  weight  13 
cwt.  3  qrs.  11  lbs.,  and  the  power  94  lbs.,  what  must 
be  the  distance  of  the  fulcrum  from  the  weight  in 
order  that  the  weight  and  power  may  balance? 

115.  If  the  longer  arm  of  a  lever  be  10  feet  and 
the  shorter  arm  2  feet  in  length,  how  must  480 
pounds  be  divided  so  that  one  part  shall  be  the 
weight  and  the  other  the  power  that  will  balance 
it  on  the  lever? 

116.  Divide  f  of  l  of  16i  by  18£  times  f  of  ff 

117.  Reduce  1  pk.  3  qts.  1  pt.  1  gill,  to  the  deci- 
mal of  a  bushel. 

118.  How  many  shillings  and  pence  in  .4562  of 
a  £? 

119.  A  general  drew  up  his  army  in  a  square,  with 
the  number  in  rank  and  file  equal,  and  had  576  men 
left.  'He  then  increased  the  square  by  placing  two 
lines  of  men  in  front,  and  two  files  on  one  side  from 
front  to  rear,  when  he  found  he  wanted  12  men  to 
complete   the  square.     How  many  men  had  he  ? 


288  MISCELLANEOUS    EXAMPLES. 

120.  What  number  is  that  one  third  of  which  ex- 
ceeds two  sevenths  of  it  by  19? 

54  9 

121.  What  number  is  that  -~  of  which  exceeds  ™ 
of  it  by  31?  28 

122.  How  many  tiles,  each  8  inches  square,  will  it 
require  to  cover  one  acre  ? 

123.  If  the  hypotenuse  of  a  right-angled  triangle 
measure  34  feet,  and  the  base  19^,  what  is  the 
measure  of  the  perpendicular? 

124.  If  the  sum  of  the  base  and  hypotenuse  is  63 
feet,  and  the  perpendicular  14  feet,  how  long  is  the 
base? 

125.  A  man  travels  south  20  miles,  then  east  15 
miles,  then  south  2J-  miles,  and  east  7  miles.  How 
far  is  he  then  from  where  he  set  out  ? 

126.  What  is  the  dhTerence  between  the  cube  root 
of  one  third  of  12,  and  one  third  of  the  cube  root 
of  12? 

127.  What  is  the  value,  in  dollars  and  cents,  of 
£94  16s.  3d.  +  £43  19s.  7d.  +  £14  13s.  9d.? 

128.  If  a  note  of  $1000,  promising  annual  interest, 
is  renewed  at  the  end  of  each  year  for  five  years, 
without  the  payment  of  any  interest,  what  is  the 
amount,  principal,  and  interest,  at  the  end  of  the 
fifth  year  ? 

129.  What  is  the  dfrTerence,  in  avoirdupois  weight, 
between  a  ball  of  silver  and  a  ball  of  gold,  each  3 
inches  in  diameter? 

130.  There  are  three  numbers;  the  first  plus  4  is 
equal  to  one  sixth  of  the  second,  the  second  is  one 
half  as  great  as  the  third,  and  the  sum  of  the  three  is 
186.     What  are  the  numbers? 

131.  There  are  two  numbers;  the  first  increased 
by  4  equals  one  sixth  of  the  second,  and  the  second 
diminished  by  6  is  eight  times  the  first.  What  are 
the  numbers? 


MISCELLANEOUS     EXAMPLES.  289 

132.  What  is  the  value,  in  Federal  money,  of  13 
shares  of  bank  stock,  par  value  $125  per  share,  and 
sold  at  11£  per  cent,  advance? 

133.  What  is  the  square  root  of  14734? 

134.  What  is  the  interest  of  $1974.36,  for  3  yrs. 

2  mo.   17  days,  at  5f-  per  cent.  ? 

135.  What  is  the  14th  term  of  a  geometrical  se- 
ries, the  first  term  of  which  is  4  and  the  ratio  f  ? 

136.  What  is  the  16th  term  of  a  geometrical  se- 
ries, the  first  term  of  which  is  7  and  the  ratio   H? 

137.  A  sells  to  B  5  loads  of  wood,  measuring,  first, 
8  ft.  6  in.,  4  ft.,  3  ft.  9  in.  ;  second,  9  ft.,  4  ft.  2  in., 

3  ft.  11  in.;  third,  9  ft.  2  in.,  4  ft.  4  in.,  4  ft.  1  in.  ; 
fourth,  8  ft.  2  in.,  3  ft.  7  in.,  4  ft.  2  in.  ;  "fifth,  7  ft. 
11  in.,  4  ft.,  3  ft.  6  in.;  at  $4.75  per  cord.  He  re- 
ceives in  payment  47  bushels  of  oats,  at  38  cents  per 
bushel  ;  56  lbs.  of  cheese,  at  8}  cents  per  pound  ; 
and  the  balance  in  butter,  at  154-  cents  per  pound. 
How  much  butter  does  he  receive? 

138.  What  is  the  weight  of  one  rod  of  lead  pipe 
one  fourth  of  an  inch  in  thickness,  if  the  inner  diame- 
ter measures  l£  inches  ? 

139.  What  is  the  weight  of  a  plate  of  iron  half  an 
inch  in  thickness,  4  feet  long,  and  2  feet  3  inches  in 
breadth  ? 

140.  How  many  feet  of  silver  wire,  one  tenth  of  an 
inch  in  diameter,  can  be  made  from  one  pound  avoir- 
dupois of  silver? 

141.  How  •many  gallons,  imperial  measure,  will  a 
cylindrical  cistern  hold,  3  feet  in  diameter  and  4^ 
feet  deep? 

142.  What  is  the  cost  of  transporting  64  barrels  of 
flour,  each  containing  7  qrs.  gross,  100  miles,  at  $3.12£ 
per  ton,  allowing  for  the  weight  of  each  cask  16  lbs."? 

143.  If  freight  by  railroad  is  $3.12^-  per  ton,  for 
100  miles,  and  freight  by  wagon  road  is  $20  per  ton 
for  80  miles,  how  much  is  saved  in,  the  freight  of  a 

25  T 


290  MISCELLANEOUS     EXAMPLES. 

barrel    of   flour   100    miles   by  railroa^   allowing    its 
weight  to  be  as  in  the  preceding  example? 

144.  At  the  rate  named  above,  how  far  could  a 
barrel  of  flour  be  carried  by  wagon  road,  before  the 
freight  should  amount  to  as  much  as  the  flour  was 
worth,  when  the  price  is   $5.62  per  barrel? 

145.  If  the  distance  from  New  York  to  Liverpool 
is  3000  miles,  what  would  be  the  cost  of  trans- 
porting a  barrel  of  flour  that  distance,  at  the  rate 
of  $20  per  ton  for  every  80  miles  ? 

146.  If  a  barrel  of  flour  can  be  transported  from 
New  York  to  Liverpool  for  65  cents,  what  would  that 
give  for  the  transport  of  one  ton  80  miles  ? 

147.  The  summit  of  the  Rocky  Mountains,  visited 
by  Freemont,  is  in  longitude  110°  8'.  What  time  is 
it  at  Greenwich  when  it  is  noon  there  ? 

148.  There  is  a  field  20  rods  long  and  8  rods  broad, 
with  a  path  3£  feet  wide  running  round  it.  How 
many  square  feet  are  there  in  the  path  ? 

149.  What  is  the  cost  of  excavating  a  cubical  pit, 
measuring  7-£  feet  in  each  direction,  at  31^  cents  per 
cubic  yard  ? 

150.  What  is  the  weight  of  an  iron  cannon  94-  feet 
in  length,  26  inches  in  diameter  at  the  larger  end,  and 
18  inches  at  the  smaller,  with  a  bore  9  feet  long  and 
10  inches  in  diameter,  allowing  for  no  inequalities  in 
the  surface  ? 

151.  What  will  be  the  duties  on  an  invoice  of  goods 
amounting  to  $1156.80,  at  30  per  cent.?. 

152.  Bought  an  invoice  of  imported  goods,  amount- 
ing to  £564  15  s.  I  agree  to  give  18  per  cent,  in  ad- 
vance of  the  invoiced  price.  What  is  the  amount 
paid,  in  Federal  money,  reckoning  $4,444  to  the 
pound  ? 

153.  I  buy  for  cash  an  invoice  of  imported  goods, 
amounting  to  £1146  16s.    Wrhat  is  the  invoiced,  price, 


MISCELLANEOUS    EXAMPLES.  291 

in  Federal  money,  allowing  Sterling  money  to  be  9 
per  cent,  in  advance  of  the  nominal  par  value  ? 

154.  A  merchant  owes  $16472.50.  He  fails,  his 
means  of  payment  amounting  to  only  $4345.62. 
How  much  is  a  creditor  entitled  to,  who  holds  a  note 
against  him  of  $  100,  dated  7i  months  previous  to  the 
final  settlement,  and  promising  interest  60  days  after 
date? 

155.  A  road,  3£  rods  wide,  is  laid  out  1  mile  and 
14£  rods  in  length,  for  which  damages  are  awarded  to 
the  land-owners,  as  follows  : — for  80  rods  of  the  road, 
at  the  rate  of  37  dollars  per  acre;  for  110  rods,  at  the 
rate  of  22  dollars  per  acre ;  and  for  the  remainder,  at 
the  rate  of  30£  dollars  per  acre.  What  is  the  whole 
amount  of  the  damages  ? 

156.  What  is  due  for  the  freight  of  12  barrels  of 
flour  65  miles,  at  $  of  a  cent  per  pound,  allowing  each 
barrel  to  contain  7  qrs.  gross,  of  flour,  and  the  cask  to 
weigh  18  pounds? 

157.  If  5  barrels  of  flour  suffice  for  a  family  of  11 
persons  7  months,  how  many  barrels  will  suffice  for  a 
family  of  15  persons  4£  months  ? 

158.  13£  times  14|  is  7£  times  what  number  ? 

159.  384  is  -±  of  how  many  times  15}  ? 

160.  Reduce  62£  cents  to  the  decimal  of  a  £,  at 
nominal  par  value. 

161.  How  many  seconds  were  there  in  the  year 
1844? 

162.  The  report  of  a  signal  gun,  fired  on  the  equa- 
tor, at  12  o'clock,  is  heard  at  a  place  due  west  distant 
16  miles.  At  what  time  is  the  report  heard  at  the 
latter  place,  allowing  69£  miles  to  a  degree  of  longi- 
tude, and  sound  to  move  1  mile  and  10  rods  in  5 
seconds  ? 

163.  A  communication  is  made  by  the  magnetic 
telegraph  from  Boston  to  Washington,  at  1  o'clock, 


292  MISCELLANEOUS     EXAMPLES. 

P.  M.  At  what  time  will  it  be  received  at  Washing- 
ton, allowing  no  time  for  the  transmission  of  the  fluid; 
longitude  of  Boston  being  71°  4'  9" ;  that  of  Wash- 
ington, 77°  T24"? 

164.  A,  engaging  in  partnership  with  B  and  C,  puts 
in  1600  dollars  for  9£  months  ;  B  puts  in  3100  dollars 
for  14  months ;  C  puts  in  2200  dollars  for  12  months. 
They  gain  1046  dollars.  What  is  each  one's  share  of 
the  gain  ? 

165.  If  100  dollars  in  one  year  gain  6£  dollars  in- 
terest, what  will  467  dollars  gain  in  9J-  months,  at  the 
same  rate  ? 

166.  What  is  the  value  of  17  shares  of  bank  stock, 
par  value  60  dollars  per  share,  and  sold  at  6£  per  cent, 
advance  ? 

167.  A  man  buys  640  barrels  of  flour  at  $5f  per 
barrel ;  pays  for  freight,  $42.75';  for  storage,  $11.17; 
and  sells  it  for  6T\  per  barrel,  allowing  a  commission 
of  2i  per  cent.     What  was  his  loss  or  gain  per  cent.  ? 

168.  The  pipe  of  an  aqueduct,  12  inches  in  diame- 
ter, is  divided  into  two  branches,  such  that  their  united 
capacity  is  equal  to  that  of  the  main  pipe,  and  the /di- 
ameter of  one  of  the  branches  is  9  inches.  What  is 
the  diameter  of  the  other  ? 

169.  The  pipe  of  an  aqueduct  is  3  feet  in  diameter. 
Wliat  number  of  pipes,  each  2£  inches  in  diameter, 
would  have  a  capacity  equal  to  that  of  the  main  pipe  ? 

170.  If  a  tree  measures,  at  the  distance  of  2  feet 
from  the  ground,  12  feet  in  circumference ;  and,  at  14 
feet  from  the  ground,  divides  into  two  branches,  meas- 
uring respectively  9  feet  and  7  feet  in  circumference ; 
how  many  square  inches  more  of  surface  would  the 
lower  horizontal  section  of  the  tree  contain  than  the 
upper  one  ? 

171.  A  certain  tree  measures,  at  the  distance  of  3 
feet  from  the  ground,  22|  feet  in  circumference.  At 
some  distauce  above,  its  four    branches  measure,  re- 


MISCELLANEOUS    EXAMPLES.  293 

spectively,  8  ft.  4  in.,  9  ft.,  7  ft.  6  in.,  and  6  ft.  5  in.  in 
circumference.  What  is  the  ratio  of  the  magnitude 
of  the  tree  at  the  lower,  compared  with  its  magnitude 
at  the  higher  place"  of  measurement  ? 

172.  The  shadow  of  a  certain  tree,  as  cast  upon  the 
ground,  measures  102  feet  in  diameter.  Allowing  the 
shadow  to  be  circular,  how  many  rods  of  ground  does 
it  cover  ? 

173.  How  many  cubic  inches  does  a  wine-glass 
contain,  measuring  3£  inches  in  depth,  and  2  inches  in 
diameter  at  the  top,  the  form  being  that  of  an  inverted 
cone  ? 

174.  How  many  yards  of  lining,  f  of  a  yard  wide, 
will  line  37£  yards  of  cloth  1^  yards  wide  ? 

175.  What  is  the  3d  term  of  the  square  900  +  420 

+  n? 

176.  What  is  the  3d  term  of  the  square  1600  +  480 

+  □? 

177.  Complete  the  square  4900  +  CD+64. 

178.  What  is  the  4th  term  of  the  cube  27000  + 
5400  +  360 +  D? 

179.  What  is  the  4th  term  of  the  cube  64000  + 
4800+120  +  □? 

180.  What  is  the  4th  term  of  the  cube  125000  + 
22500+1350  +  □? 

181.  What  is  the  cube  root  of  68437? 

182.  What  is  the  cube  root  of  954326  ? 

183.  What  is  the  solid  contents  of  the  largest  sphere 
that  can  be  cut  from  a  cubic  block  13£  inches  on  a 
side  ? 

184.  From  a  sphere  measuring  10  inches  in  diame- 
ter, the  largest  possible  cubic  block  has  been  cut ;  and 
from  this  block  again  the  largest  possible  sphere  has 
been  cut.  What  is  the  diameter  of  the  last-named 
sphere  ? 

185.  From  a  sphere  20  inches  in  diameter  what  are 
the  dimensions  of  the  largest  parallel  prism  that  can 

25* 


294      '  MISCELLANEOUS    EXAMPLES. 

be  cut,  making  the  length  to  the  breadth  as  4  to  3,  and 
the  breadth  to  the  thickness  as  3  to  2  ? 

186.  How  many  cubic  feet  are  there  in  the  Avails  of 
a  brick  house,  the  length  and  breadth  of  which  are 
each  44  feet  outside,  and  the  height  24  feet,  supposing 
the  walls  to  be  perpendicular  outside,  and  the  thickness 
to  be  2  feet  at  the  bottom  and  1  foot  at  the  top,  making 
no  deduction  for  doors  or  windows  ? 

187.  What  are  the  prime  factors  of  3746?  Of  9862  ? 

188.  What  is  the  least  common  multiple  of  684, 
963.  and  8416? 

189.  What  is  the  greatest  common  divisor  of  94620, 
3642,  and  1646  ? 

190.  How  many  times  will  the  wheel  of  a  railroad 
car,  if  it  be  24-  feet  in  diameter,  revolve  in  going  40 
miles  ? 

191.  How  many  times  will  such  a  wheel  revolve 
in  a  minute,  if  the  speed  of  the  car  be  20  miles  an 
hour  ? 

192.  What  is  the  cost,  in  Federal  money,  of  the 
freight  of  940  bales  of  cotton,  averaging  340  lbs.  per 
bale,  at  f  of  a  penny  per  pound? 

193.  Sold  34  pieces  cotton  goods,  averaging  31^ 
yards  each  in  length,  at  9|  cents  per  yard,  l£  per  cent, 
off.     What  is  the  amount  of  the  bill  ? 

194.  Bought  840  barrels  of  flour,  on  six  months' 
credit,  at  4&  dollars  per  barrel.  Sold  the  flour  the 
same  day  on  three  months'  credit,  at  4j|  dollars  per 
barrel.  Did  I  gain  or  lose,  and  how  much,  estimating 
the  present  worth  of  the  debts  at  the  date  of  the 
transaction  ? 

195.  A  merchant  buys  for  me,  on  commission,  400 
barrels  of  flour,  for  $4.34  per  barrel,  cash.  He  sells  the 
flour  on  the  same  day  for  cash,  at  $4.46  per  barrel. 
How  much  do  I  gain  by  the  operation,  allowing  1£  per 
cent,  commission  on  the  purchases,  and  2  per  cent,  on 
the  sales? 


MISCELLANEOUS     EXAMPLES.  295 

196.  A  sets  out  on  a  journey,  travelling  24  miles  a 
day.  B  sets  out  2  days  after,  travelling  3 If  miles  a 
day.  A,  after  travelling  3  days,  goes  25  miles  a  day ; 
and  B,  after  travelling  4  days,  goes  33£  miles  a  day. 
In  how  many  days  after  A  sets  out  will  B  overtake 
him  ? 

197.  If  a  parallel  prism  is  4£  inches  thick,  5  inches 
wide,  and  6  inches  long,  what  must  be  the  diameter 
of  the  hollow  sphere  that  would  enclose  it  ? 

198.  If  18  horses,  in  16  weeks,  consume  204  bush- 
els of  oats,  how  many  horses  will  it  require  to  consume 
413  bushels  in  18  weeks  ? 

199.  If  100  dollars  gain  6  dollars  interest  in  12 
months,  what  will  be  the  interest  of  840  dollars  for 
5}  months  ? 

200.  If  a  field  containing  17  acres  measures  81  rods 
on  one  side,  what  must  be  the  length  of  the  corre- 
sponding side  of  a  similar  field  containing  26J-  acres  ? 

201.  The  contents  of  two  similar  fields  are  as  4j- 
to  7,  and  the  smaller  measures  on  one  side  63  rods. 
What  must  be  the  corresponding  dimension  of  the 
larger  field  ? 

202.  There  are  two  circles ;  their  areas  are  as  14  to 
19,  and  the  diameter  of  the  smaller  is  16  rods.  What 
is  the  diameter  of  the  larger  ? 

203.  What  is  the  area  of  a  circle  whose  diameter  is 
44  feet  ? 

204.  What  is  the  circumference  of  a  circle  whose 
diameter  is  67  inches  ? 

205.  Given  the  circumference  of  a  circle  60  rods,  to 
find  its  area. 

206.  There  are  three  equilateral  triangles,  whose 
areas  are  to  each  other  as  the  numbers  3,  4,  and  7.  A 
side  of  the  smallest  measures  40  rods.  What  is  the 
sum  of  their  areas  ? 

207.  A  grindstone  is  3  feet  in  diameter.  Allowing 
the  hole  in  the  middle  to  be  2  inches  in  diameter,  how 


296  MISCELLANEOUS    EXAMPLES. 

many  inches  must  be  ground  off  to  grind  away  half 
of  the  stone  ? 

208.  The  fore  wheels  of  a  wagon  are  3  feet  10 
inches  in  diameter,  and  the  hind  wheels  4  feet  2 
inches  in  diameter.  How  many  times  more  does  one 
of  the  fore  wheels  turn  round  than  one  of  the  hind 
wheels  in  going  one  mile  ? 

209.  What  is  the  weight  of  a  cast-iron  cylinder  6 
feet  long  and  4i  inches  in  diameter,  the  specific  grav- 
ity being  as  already  given? 

210.  The  frustum  of  a  cone  7  feet  long  is  14  inches 
in  diameter  at  the  larger  end,  and  10  inches  in  diam- 
eter at  the  smaller.  How  far  from  the  base  must  it 
be  cut  in  two  to  divide  it  into  equal  parts  ? 

211.  What  is  the  first  prime  number  above  901  ? 

212.  What  is  the  first  prime  number  below  10000  ? 

213.  What  is  the  greatest  common  divisor  of  1846, 
3105,  684,  and  1006? 

214.  What  are  all  the  prime  factors  of  801,  of  3042, 
of  586,  of  908  ? 

215.  In  what  proportion  may  corn  at  80  cents  be 
mixed  with  rye  at  86  cents,  and  with  oats  at  43  cents, 
per  bushel,  to  make  the  mixture  worth  50  cents  per 
bushel  ? 

216.  Add  the  fractions  v+^  +  ^7- 

217.  What  is  the  value  of  ?  of  ££,  f  of  T\s.,  ex- 
pressed in  the  decimal  of  a  £  ? 

218.  What  is  the  present  value  of  a  note  of  584 
dollars,  payable  in  three  months? 

219.  What  is  the  bank  discount  on  a  note  of  150 
dollars,  payable  in  three  months? 

220.  What  sum  will  be  paid  on  a  note  of  240  dol- 
lars, discounted  at  a  bank,  for  90  days  ? 

221.  What  is  the  interest  of  1200  dollars  for  10 
days,  at  7  per  cent.  ? 


MISCELLANEOUS     EXAMPLES. 


297 


222.  Divide  the  sum  of  the  decimals  2016  +  9172 
-f-  0064,  by  £  of  f  reduced  to  a  decimal. 

223.  Divide  3  T.  17  cwt.  3  qrs.  19  lbs.  by  6. 

224.  Divide  9  m.  3  fur.  21  r.  14  ft.  by  8. 

225.  Multiply  31  d.  14  h.  37  m.  15  sec.  by  19. 

226.  Multiply  83  A.  3  R.  22  r.  by  12. 

227.  What  is  one  fifth  of  16  T.  11  cwt.  2  qrs. 
20  lbs.  ? 

228.  What  is  the  value,  at  4  dollars  a  cord,  of  three 
loads  of  wood,  measuring  as  follows  :  first,  8  feet  6  in., 
4  ft.  2  in.,  3  ft.  9  in.  ;  second,  9  ft.,  4  ft.  1  in.,  3  ft. 
10  in.  ;  third,  8  ft.  1  in.,  4  ft.,  4  ft.  2  in.  ? 

229.  What  is  the  15th  term  of  an  arithmetical 
series,  the  1st  term  of.  which  is  3  and  the  common 
diri'erence  §  ? 

230.  If  the  9th  term  of  an  arithmetical  series  is  23, 
and  the  common  difference  |,  what  is  the  3d  term  ? 

231.  What  is  the  sum  of  an  arithmetical  series  of 
40  terms,  if  the  1st  term  is  2  and  the  common  dif- 
ference 3£? 

232.  What  is  the  sum  of  an  arithmetical  series  of 
72  terms,  if  the  1st  term  is  1  and  the  common  dif- 
ference If  ? 

233.  A  man  engages  to  walk  1000  miles  in  1000 
hours,  on  condition  of  receiving  1  cent  for  the  first 
mile,  and  for  each  mile  after  £  of  a  cent  more  than  he 
had  for  the  mile  preceding  it..  What  will  he  be  enti- 
tled to  on  the  fulfilment  of  his  contract  ? 

234.  There  are  two  grindstones,  the  thickness  of 
which  is  to  the  diameter  as  2  to  11.  The  smaller 
one  is  2  feet  in  diameter  ;  the  other  is  three  times  as 
heavy.     What  is  its  diameter  ? 

235.  A  man  has  a  triangular  field  containing  7  acres. 
The  vertex  or  point  of  the  triangle  is  54  rods  from  the 
base.  At  what  distance  from  the  base  must  a  line  be 
drawn  parallel  to  it  so  as  to  cut  off  one  half  the  field  ? 

236.  The  hypotenuse  and    perpendicular  of  a  tri- 


298  MISCELLANEOUS    EXAMPLES. 

angle  measure  together  816  rods ;  the  base  measures 
61  rods.     What  is  the  length  of  the  perpendicular? 

237.  A  rope  100  feet  long  passes  straight  from  the 
ground,  at  the  distance  of  10  feet  from  a  perpendicu- 
lar pole,  over  the  top  of  the  pole,  which  is  25  feet  in 
height,  and  thence  is  drawn  so  as  to  reach  the  ground 
at  the  farthest  possible  point.  Allowing  the  ground 
to  be  level,  how  far  is  the  last-named  point  from  the 
foot  of  the  pole  ? 

238.  A  stick  of  timber,  in  the  form  of  a  truncated 
wedge,  is  10  feet  long,  2  feet  wide  through  its  whole 
extent,  20  inches  in  thickness  at  one  end,  and  14 
inches  thick  at  the  other.  How  far  from  the  thicker 
end  must  it  be  cut  in  two  so  as  to  divide  it  into  two 
equal  parts  ?  .    * 


RECOMMENDATIONS. 


From  Mr.  George  B.  Emerson,  Boston. 


I  have  carefully  examined  the  plan  of  Mr.  Adams's  work  on  Men- 
tal Arithmetic,  and  have  given  some  attention  to  its  execution  ;  and  I 
am  confident  that  it  will  prove  a  very  valuable  addition  to  the  means 
of  instruction  in  Arithmetic.  It  is  a  successful  extension  of  the  ad- 
mirable method  of  Colburn's  First  Lessons,  with  such  modifications 
as  seemed  to  be  required  in  a  higher  work  on  the  same  general 
model.  It  occupies  unappropriated  ground  ;  and  it  deserves,  and  I 
think  it  will  take,  a  high  place  amongst  the  text-books. 

GEO.    B.   EMERSON. 


From  Mr.  Thomas  Sherwin,  Boston. 


I  have  carefully  examined,  in  manuscript,  the  work  of  Mr.  Adams 
on  Mental  Arithmetic,  and  am  much  pleased  with  it.  His  plan  is 
good,  and  well  executed.  I  would,  therefore,  heartily  recommend 
his  book  to  Teachers  and  School  Committees,  as  one  which  will  con- 
tribute very  materially  to  the  attainment  of  that  very  important,  but 
much-neglected,  branch  of  study,  —  Intellectual  Arithmetic. 

THOMAS   SHERWIN, 
Principal  of  the  Boston  English  High  School. 


From  Mr.  Solomon  Adams,  Boston. 

To  the  Publisher. 

Dear  Sir  :  —  Having  been  favored  with  an  opportunity  of  exam- 
ining, in  manuscript,  a  Treatise  on  Mental  Arithmetic,  by  F.  A. 
Adams,  A.  M.,  I  am  most  happy  to  find  that  our  schools  are  about  to 
have  a  work  of  the  kind,  carried  with  much  skill  and  judgment  into 
the  higher  departments  of  Arithmetic. 

Very  respectfully  yours, 

SOLOMON   ADAMS, 
Principal  of  the  Young  Ladies"  School,  Central  Place. 


RECOMMENDATION'S.  O 

From  Roger  S.  Howard,  Esq.,  Newhuryport. 

Mr.  F.  A.  Adams. 

Dear  Sir:  —  I  have  looked  over,  with  much  care  and  pleasure,  the 
manuscript  Arithmetic,  which  you  put  into  my  hands  a  few  days  since. 
The  plan  of  the  work  appears  to  me  quite  original,  and  many  of  the  me- 
thods you  have  adopted  exceedingly  ingenious,  and,  at  the  same  time, 
beautifully  simple.  Your  rules  and  explanations  are  clear  and  concise ; 
and  the  numerous  examples  for  practice  which  you  have  inserted,  are 
judiciously  selected  and  well  arranged.  The  book,  I  think,  is  one  which 
will  greatly  facilitate  the  teaching  of  this  important  branch  of  education. 
I  am,  sir,  very  respectfully  yours, 

ROGER  S.  HOWARD, 
Principal  of  the  Putnam  High  School. 


From  Mr.  Rufus  Putnayn,  Salem. 

Mr.  F.  A.  Adams. 

Dear  Sir:  —  I  have  read  with  much  satisfaction  the  manuscript  copy 
of  the  Mental  Arithmetic  you  are  intending  to  publish.  The  plan  of  the 
work  is,  in  many  respects,  different  from  its  predecessors ;  and,  strange 
as  it  may  seem,  to  those  who  examine  many  of  the  new  books  in  the 
various  departments  of  education,  and  who  have  not  read  yours,  it  occu- 
pies much  ground  which  has  not  been  occupied  by  others.  I  think  that, 
in  its  arrangement,  its  definitions,  its  explanations,  the  examples  for 
practice, —  indeed,  in  its  whole  matter,  —  it  is  happily  adapted  to  its  ob- 
ject ;  to  release  our  youth  from  a  part  of  their  present  bondage  to  slate 
and  pencil,  and  artificial  rules,  by  qualifying  them  to  perform  correctly 
and  easily,  in  the  mind,  many  of  the  operations  which  are  almost  univer- 
sally performed  on  the  slate.  I  commend  it,  with  much  confidence,  to  the 
notice  of  all  who  are  intrusted  with  the  education  of  youth. 
Yours,  verv  truly  and  respectfully, 

R.  PUTNAM, 
Principal  of  the  Bowditch  (English  High)  School. 


From  Mr.  Edwin  Jocelyn,  Salem. 
Mr.  F.  A.  Adams. 

Dear  Sir:  —  No  one  can  hold  "Colburn's  First  Lessons  in  Arithme- 
tic" in  higher  estimation  than  I  do;  and  I  think,  whoever  undertakes  to 
furnish  a  substitute  for  that  little  book,  which  shall  better  answer  the 
purpose,  will  fail  in  his  purpose.  I  am  glad  to  see  from  your  hand  an  ex- 
tension of  Mental  Arithmetic  on  the  plan  of  that  inestimable  school-book. 
I  have  often  felt  the  want  of  such  a  work,  and  have  in  practice  extended 
this  course  of  teaching,  somewhat :  —  and  should  have  done  it  oftener 
and  farther,  if  I  had  had  such  a  book  at  hand  as  you  now  propose  to  pub- 
lish. The  plan  appears  to  me  to  be  very  happily  carried  out,  and  I  feel 
confident  that  it  will  meet  with  a  wide  appreciation  and  use. 
Yours,  with  much  esteem, 

EDWIN  JOCELYN, 
Principal  of  the  Female  High  School. 


4  RECOMMENDATIONS. 

From  Mr.  Charles  Norihend,  Salem. 
Mr.  F.  A.  Adams. 

Dear  Sir:  —  Having  examined,  with  some  care,  your  manuscript  en- 
titled "Advanced  Lessons  in  Mental  Arithmetic,"  I  feel  no  hesitation  in 
saying,  that  I  consider  the  work  very  happy  in  design,  and  admirable  in 
execution.     Wishing  you  much  success, 

I  remain  very  truly  yours,  &c, 

CHARLES  NORTHEND, 
Principal  of  Epes  Grammar  School. 


From  the  North  American  Review. 

To  the  late  Warren  Colburn  belongs  the  high  credit  of  first  introducing 
into  our  Schools,  through  his  admirable  First  Lessons,  the  regular  study 
of  Mental  Arithmetic.  Of  this  excellent  little  manual,  the  author  of  the 
book  before  us  justly  observes,  that,  so  completely  has  it  performed  the 
work  within  its  prescribed  sphere,  that  there  is  little  reason  to  desire  or 
to  expect  that  it  will  ever  be  superseded.  Mr.  Colburn  published  also  a 
Sequel  to  Mental  Arithmetic,  in  which  the  principles  and  rules  of  Written 
Arithmetic  were  deduced  from  the  solution  and  analysis  of  questions  ac- 
cording to  the  method  adopted  in  the  former  treatise.  This  Sequel  was 
very  well  executed  as  far  as  it  went ;  but  it  was  not  full  enough  for  all 
the  wants  of  the  higher  classes  in  our  schools.  It  omitted  Proportion  and 
Progression,  the  "Rule  of  Three,"  and  the  doctrine  of  Powers  and  Roots. 
Mr.  Adams  has  undertaken  to  supply  these  deficiencies,  following  mainly 
in  the  track  of  Mr.  Colburn,  but  appearing  fully  competent  also  to  mark 
out  a  path  for  himself.  By  this  enlargement  of  plan,  he  has  brought 
many  useful  problems  in  Mensuration  and  Mechanics  within  the  scope 
of  his  work,  and  has  extended  the  analysis  and  induction  over  much  new 
ground,  though  many  of  the  new  problems  are  still  left  to  be  performed 
by  arbitrary  rules. 

The  First  Part  of  Mr.  Adams's  book  consists  of  exercises  in  Mental 
Arithmetic,  arranged  under  the  different  arithmetical  rules.  Where  the 
principles  have  not  been  taught  in  the  First  Lessons,  they  are  here  care- 
fully deduced  from  an  analysis  of  a  number  of  simple  questions,  following 
which  are  numerous  and  well-selected  examples.  These  examples  pass 
gradually  from  simple  to  more  complicated  questions,  so  as  to  give  the 
pupil  a  thorough  training.  In  the  Second  Part,  the  different  processes 
are  arranged  in  the  same  order  as  before ;  and  when  the  operations  are 
complicated,  distinct  rules  are  given,  illustrated  by  examples  for  practice 
containing  larger  numbers  than  were  suitable  for  the  exclusively  mental 
operation.  When  the  operations  are  simple,  and  sufficiently  explained  in 
the  analysis  and  induction  contained  in  the  First  Part,  a  reference  is 
merely  made  to  that  Part,  and  the  examples  for  practice  follow,  without 
any  enunciation  of  a  rule. 

The  author's  reasoning  and  explanations  are  very  clear,  simple,  and 
concise ;  his  disposition  of  the  different  parts  judicious,  and  his  selection 
of  examples  well  suited  to  exercise  the  mind  of  the  pupil.  As  a  whole, 
we  prefer  this  work  to  any  Arithmetic  we  have  seen  in  use. 


RECOMMENDATIONS.  O 

From  Professor  Chase,  of  Dartmouth  College. 
Mr.  F.  A.  Adams. 

My  Dear  Sir  :  —  I  have  examined,  with  some  care,  your  Treatise  on 
Arithmetic,  and  am  much  pleased  with  it.  The  practice  and  habit  of 
extending  mental  operations  to  large  numbers,  is  of  great  utility.  I  have 
occasion,  very  frequently,  to  see  the  inconvenience  that  young  men  suffer, 
from  the  want  of  such  a  habit.  Not  less  valuable  than  the  habit  of  ope- 
rating mentally  upon  large  numbers,  is  the  habit  of  performing  the  more 
advanced  operations  of  arithmetic  without  the  aid  of  the  pencil. 

I  like  very  much,  also,  the  manner  in  which  you  have  treated  several 
of  the  principles  which  you  have  developed ;  as,  for  example,  the  subject 
of  the  Common  Divisor,  the  Least  Common  Multiple,  the  Roots,  Ratio, 
and  Proportion.  These  are  but  few  of  the  subjects,  but  I  mention  them 
as  examples. 

I  think  the  book  will  do  much  to  promote  the  proper  method  of  teach- 
ing arithmetic,  —  by  demonstration  and  explanation. 

I  am,  dear  sir,  very  truly  yours,  &c, 

S.  CHASE. 


From  Mr.  Addison  Brown,  of  Brattleboro,  Vt. 
Daniel  Bixby,  Esq. 

Dear  Sir  :  —  I  give  you  many  thanks  for  the  "  First  Book  in  Arithme- 
tic," by  F.  A.  Adams,  which  you  had  the  kindness  to  send  me,  some  time 
since.  To  test  its  merits,  I  set  my  youngest  child,  a  daughter,  seven 
years  of  age,  to  studying  it.  She  has  been  about  half  through  it ;  and 
having  heard  her  lessons  myself,  and  watched  her  progress,  and  the  effect 
the  book  has  had  in  developing  her  powers  of  calculation,  I  am  satisfied 
that  it  is  a  very  excellent  work.  I  think,  as  a  First  Book  in  Arithmetic, 
it  is  decidedly  the  best  I  have  either  used  or  examined,  and  I  should  be 
glad  to  see  it  extensively  introduced  into  our  Schools. 

Yours,  with  respect, 

ADDISON  BROWN. 


From  Mr.  John  Tatlock,  Professor  of  Mathematics,  and  Mr.  A.  Hopkins, 
Professor  of  Natural  Philosophy. 

I  have  examined  a  treatise  on  Arithmetic,  by  F.  A.  Adams,  and  am 
much  pleased  with  it.  I  think  it  well  adapted  to  teach  the  science  and 
art  of  numbers,  and  at  the  same  time  to  teach  the  art  of  thinking.  I  am 
persuaded  that  a  thorough  training  in  this  Arithmetic  would  prepare 
students  for  the  further  study  of  mathematics  better  than  nine-tenths  are 
now  prepared. 

I  should  be  glad  if  every  student  who  enters  college  was  master  of 
this  Arithmetic. 

JOHN  TATLOCK. 
A.  HOPKINS. 


D  RECOMMENDATIONS. 

From  Mr.  Roger  B.  Hildreth,  Tyngshorough. 
To  the  Publisher. 

Dear  Sir  :  —  The  "  Advanced  Lessons  in  Mental  Arithmetic,"  upon 
the  plan  adopted  by  Mr.  F.  A.  Adams,  will,  I  am  persuaded,  from  a  care- 
ful  examination  of  the  work,  be  found  very  useful  in  instruction.  It 
peculiarly  meets  the  wants  both  of  the  teacher  and  the  pupil ;  and  as  its 
real  merits  become  more  generally  known,  I  am  confident  it  will  be  ex- 
tensively used  as  a  text-book.  The  work  is  arranged  with  admirable 
clearness,  —  the  rules  and  explanations  are  concise,  and  yet  very  simple. 
I  cheerfully  commend  it  to  the  notice  of  all  who  wish  for  an  excellent 
treatise  on  Mental  Arithmetic. 

Very  respectfully  yours, 

ROGER  B.  HILDRETH, 
Principal  of  Tyngshorough  High  School. 


From  J.  II.  Purkitt,  Esq.,  St.  Louis. 

Mr.  Bixby. 

Dear  Sir  :  —  I  have  used  Frederic  A.  Adams's  Arithmetic  in  my 
School  for  many  months ;  and  it  gives  me  pleasure  to  say  that  it  has  met 
my  highest  expectations.  I  regard  it  as  b}'  far  the  best  Arithmetic  in 
the  English  language.  Recommendations  are  extremely  cheap,  nor 
-would  I  trouble  you  with  an  expression  of  my  own  opinion  of  its  merits, 
did  I  not  believe  it  to  be  an  act  of  common  justice,  when  an  author  has 
produced  a  really  valuable  book,  that  those  who  have  thoroughly  tested  it, 
and  know  and  feel  its  value,  should  freely  and  cordially  acknowledge  it. 
If  it  has  a  decided  superiority  over  all  other  Arithmetics,  in  developing 
power  of  thought,  and  training  the  mind  to  a  stern  and  rigid  analysis, 
then  the  public  have  a  right  to  know  it.  That  this  book,  faithfully  used, 
will  do  this,  I  sincerely  believe.  No  consideration  whatever  could  induce 
me  to  throw  it  out  of  my  school,  for  the  sake  of  introducing  any  other 
Arithmetic  yet  published.  I  feel,  therefore,  that  I  am  doing  not  only 
what  is  just,  but  what,  I  hope,  is  a  real  good,  by  this  expression  of  opi- 
nion. Teachers  may  be  slow  in  using  it  as  a  text-book  in  their  schools ; 
but  that  it  will  be  ultimately  adopted,  and,  when  adopted,  permanently 
retained,  I  cannot,  for  a  moment,  doubt.  To  the  la?y  and  unthinking, 
and  those  who  are  bigotcdly  attached  to  the  common  mode  of  teaching 
Arithmetic,  this  book  will  not  only  not  be  acceptable,  but,  perchance,  be 
dogmatically  condemned,  and  indignantly  rejected.  But  those  who  love 
to  think  themselves,  and  desire  to  develope  the  power  of  correct  thinking 
in  others,  will  find  in  this  book  an  exhibition  of  principles,  beautiful  as 
they  are  true,  and  ingenious  as  they  are  rational.  Wishing  you  great 
success, 

I  am,  dear  sir,  yours,  with  respect, 

J.  H.  PURKITT, 
Principal  of  the  Young  Ladies1  High  School. 


RECOMMENDATIONS.  1 

From  Mr.  Charles  C.  Dame,  Principal  of  the  English  High  School,  New- 
buryport. 

Mr.  F.  A.  Adams. 

Dear  Sir  :  —  I  have  examined  somewhat  carefully  your  "  Mental  and 
Written  Arithmetic,"  and  find  that  it  possesses  peculiar  merit.  The 
arrangement  and  exercises  are  such,  I  think,  as  will  train  the  student  to 
a  habit  of  thinking  and  reasoning  for  himself,  and  lead  him  readily  to 
apprehend  the  relations  of  numbers  in  their  various  combinations.  The 
book  is  not  encumbered  with  "many  rules,"  but  the  subjects  of  which  it 
treats  appear  to  be  taken  up  in  a  systematic  manner;  and  the  expla- 
nations are  clear  and  analytic. 

The  work  throughout  seems  to  have  been  planned  and  executed  in  a 
way  well  calculated  to  fit  those  who  may  use  it  for  the  "  active  pursuits 
of  life," —  to  enable  them  to  solve,  in  the  most  ready  and  natural  way, 
those  arithmetical  questions  which  may  occur  in  business  or  otherwise. 
The  questions  for  exercise  which  it  contains  are,  for  the  most  part,  prac- 
tical, and  so  constructed  as  to  afford  encouragement  in  their  solution. 

I  have  formed  such  an  opinion  of  the  work,  as  to  believe  that  no  scholar 
will  hereafter  leave  school  where  it  is  known,  and  consider  himself  well 
educated,  unless  he  is  familiar  with  its  pages,  or  the  principles  which 
they  contain. 

I  am  very  truly  and  respectfully  yours, 

CHARLES  C.  DAME. 


From  E.  Wyman,  Esq.,  St.  Louis. 
Mr.  F.  A.  Adams. 

Dear  Sir  :  —  It  is  now  nearly  two  years  since  I  introduced  your  Arith- 
metic into  my  School,  and  I  have  refrained  from  expressing  any  opinion 
of  its  merits,  until  it  should  have  been  fairly  and  thoroughly  tested.  I 
did  not  introduce  it  —  nor  do  I  any  book  —  simply  as  a  matter  of  experi- 
ment. I  believed  it  to  be  a  good  work,  from  examination  of  it ;  and  am 
now  prepared  to  add,  that  I  know  it  to  be  a  good  one,  from  trial. 

I  cheerfully  assent  to  the  opinions  I  see  expressed  by  others,  on  all  the 
valuable  characteristics  of  the  book.  Beside  these,  I  must  mention  the 
fact,  that  upon  examination  of  those  students  who  have  been  carried 
through  the  system,  and  made  to  comprehend  it,  I  find  them,  in  their 
arithmetical  processes,  independent  of  arbitrary  diction  of  rule,  possessing 
a  strong  analytical  power,  and  arriving  at  results  with  great  rapidity  and 
accuracy.  The  head-work  takes  precedence  of  the  hand- work ;  and  they 
have  a  why  and  wherefore  for  what  they  do.  My  commendation  of  the. 
book  is  full  and  unequivocal. 

Yours  truly, 

E.  WYMAN,  A.  M., 

Principal  of  the  St.  Louis  English  and  Classical  High  School. 


a  «  ' 

0  RECOMMENDAiTIOiNS. 

From  Elias  Nusan,  A.  M.,  Principal  of  the  Classical  High  School, 

Newbimjperl. 

Mr.  Adams. 

Dear  Sir  : — Having  carefully  examined  your  treatise  on  Arithmetic,  I 
would  say  that!  have  formed  a  very  favorable  opinion  of  it,  and  believe 
it  to  be  a  valuable  contribution  to  the  department  of  instruction  to  which 
it  relates.  Your  method  of  performing  mental  operations  on  large  num- 
bers is  philosophical,  and,  for  the  most  part,  original ;  )^our  rules  and  illus- 
trations are  written  with  clearness  and  precision,  and  your  examples  for 
practice  have  been  chosen  with  reference  to  the  actual  concerns  of  life. 
You  have  not  encumbered  the  book  with  arithmetical  puzzles,  which  tend 
to  perplex  the  pupil,  and  occupy  time  which  should  be  devoted  to  useful 
problems.  The  practical  and  business-like  character  of  the  work  is,  in 
my  opinion,  a  great  excellence.  The  examples  are  such  as  are  of  daily 
occurrence  in  the  family,  the  work-shop,  and  the  counting-room ;  and  the 
scholar  who  has  become  master  of  them  will  not  need  another  course  of 
training,  and  a  new  set  of  rules,  to  meet  the  actual  wants  of  trade  and 
business  in  the  world. 

Your,  method  of  working  Fractions,  both  Vulgar  and  Decimal,  is 
rational  and  easy ;  and  the  doctrines  of  Proportion  and  of  the  Roots  arc 
plainly  and  distinctly  unfolded. 

The  exposition  of  the  laws  of  Mechanics,  and  the*  sections  on  the  Com- 
parison of  Similar  Surfaces  and  Solids,  on  Per  Centage,  Mensuration,  and 
on  Weights  and  Measures,  are  exceedingly  valuable. 

In  short,  I  cannot  but  think  that  you  have  presented  the  whole  science 
of  arithmetic  in  a  very  interesting  and  philosophical  point  of  view;  and 

1  believe  that  those  teachers  who  use  your  book  wiilj  find  it  admirably 
fitted  both  to  develop  the  intellectual  powers  of  their  pupils,  and  to  make 
them  quick  and  accomplished  arithmeticians. 

With  respect,  I  remain  yours, 

ELIAS  NASON. 


From  Mr.  William  Smyth,  Professor  of  Mathematics,  Bitwdoin  College. 

I  have  examined  the  system  of  Arithmetic  by  the  Rev.  F.  A.  Adams, 
Principal  of  Dummer  Academy.  The  plan  of  the  work,  and  the  style  of 
its  execution,  appear  to  me  well  calculated  to  give  to  the  learner  clear 
views  of  the  general  principles  and  operations  of  Arithmetic,  and  to 
furnish  the  discipline  requisite  to  a  skilful  and  ready  application  of  them. 
The  work,  indeed,  as  should  be  the  casern  all  works  of  the  kind,  appears 
to  have  been  composed  in  the  recitation  room,  by  one  well  conversant 
with  his  subject,  and  possessing,  in  an  eminent  degree,  the  talents  re- 
quisite to  a  successful  instructor ;  and  is  therefore  admirably  adapted  to 
the  wants  both  of  the  pupil  and  teacher.  I  should  regard  with  much 
pleasure  its  extensive  introduction  into  our  schools  and  academies. 

WE  SMYTH.    - 


RECOMMENDATIONS.  9 

From  Mr.  John  D.  Philbrick,  Principal  of  the  Mathematical  Department 
of  the  Mayhem  School,  Boston. 

Mr.  F.  A.  Adams. 

My  Dear  Sir: —  I  am  delighted  with  your  Arithmetic.  A  careful 
examination  of  every  page  of  it  has  convinced  me  that  it  is  a  work  of 
icendcnt  excellence.  To  say  that  it  contains  a  great  amount  of  mat- 
ter well  arranged ;  that  its  rules  and  explanations  are  clear  and  logical, 
and  the  examples  well  adapted  to  illustrate  them,  would  be  to  accord  to 
it  but  a  small  part  of  its  just  meed  of  praise. 

Its  peculiar  and  crowning  merit  is,  that  it  is  calculated  to  emancipate 
the  learner  from  the  bondage  of  rules,  and  even  to  give  him  dominion 
over  them,  so  that  they  shall  be  to  him  as  clay  in  the  hands  of  the  potter. 
I  cannot  but  regard  it  as  a  superstructure  worthy  of  its  admirable  basis, 
Colburn's  First  Lessons;  and  if  the  one  be  a  "faultless"  school-book,  the 
other  is  not  a  whit  less  perfect.  I  am  confident,  therefore,  that  it  needs 
no  other  recommendations  than  its  own  merits,  to  insure  it  a  hearty  wel- 
come every  where  among  intelligent  teachers. 

Yours  truly, 

JOHN  D.  PHILBRICK. 


From  Mr.  Stephen  Holman,  Principal  of  Fitchburg  Academy. 

Mr.  F.  A.  Adams. 

Dear  Sir  :  —  I  have  examined  your  work  on  Arithmetic,  and  find  it 
admirably  adapted  to  teach  the  "  science  of  numbers."  It  will  be  re- 
ceived as  a  valuable  assistant,  by  those  teachers  who  aim  to  give  their 
pupils  a  thorough  knowledge  of  the  properties  of  numbers,  as  it  enables 
the  learner  to  derive  much  from  the  text-book,  which  has  heretofore  been 
communicated  orally,  if  at  all.  The  scholar  who  studies  this  book  faith- 
fully, will  see  the  subject  divested  of  its  mystery,  will  learn  to  worship 
rules  less,  and  common  sense  more. 

I  have  seen  no  Arithmetic  so  well  adapted  to  the  wants  of  our  schools. 
I  am,  sir,  very  respectfully  yours, 

STEPHEN  HOLMAN. 


From  Mr.  A.  K.  Hathaway,  Principal  of  the  Grammar  School,  Medford. 

Mr.  Adams.  4K 

Dear  Sir  :  —  I  have  very  carefully  examined  the  "  Advanced  Lessons 
in  Mental  Arithmetic,"  and  with  the  highest  satisfaction.  The  plan  of 
the  work  is  admirable,  and  the  execution  shows  great  care  and  judg- 
ment. 

This  work  occupies  new  and  unappropriated  ground,  and  is  just  the 
manual  most  needed  in  our  schools,  to  give  that  high  degree  of  mental 
discipline,  so  necessary  in  training  the  young.  It  needs  only  to  be  known 
to  be  appreciated ;  and,  when  once  known,  I  am  fully  confident  it  will  be 
very  extensively  used. 

Yeurs  very  truly, 

A.  K.  HATHAWAY. 


10  RECOMMENDATIONS. 

From  Rev.  James  Means,  Principal  of  Lawrence  Academy,  Groton. 
Daniel  Bixby,  Esa. 

Dear.  Sir  :  —  I  have  carefully  examined  the  "  Mental  and  Written 
Arithmetic "  which  you  put  into  my  hands,  and  am  prepared  to  give  it 
my  unqualified  approval. 

It  would  be  inappropriate  to  detail  all  the  excellences  which  it  pos- 
sesses. I  will  just  mention,  as  points  which  gratify  me,  the  method  of 
multiplying  and  dividing-  large  numbers,  used  by  all  good  accountants, 
and  now  introduced  for  the  first  time  into  a  school-book ;  the  manner  of 
stating  and  illustrating  the  rules ;  the  nature  of  the  questions  proposed 
for  solution ;  the  introduction  of  matter  quite  new  upon  several  points  of 
practical  importance.  I  would  refer  to  the  rules  for  Compound  Propor- 
tion, Square  and  Cube  Roots,  several  parts  of  Section  XLIV.,  and  all  of 
Section  XLVI.,  as  worthy  of  special  notice. 

I  hope  and  believe  that  this  book  will  be  extensively  used.  I  shall 
commend  it  to  the  Committee  of  the  Trustees  here,  who  have  such  mat- 
ters in  charge,  and  make  no  doubt  it  will  be  permanently  established  as 
a  text-book. 

Very  respectfully  yours, 

JAMES  MEANS. 


From  Teachers  of  the  Public  Schools  in  Lowell. 

Having  carefully  examined  the  Mental  and  Written  Arithmetic  by  F. 
A.  Adams,  we  do  not  hesitate  to  say,  that,  in  its  design  "  to  continue  and 
extend  the  course  of  discipline  in  numbers,"  it  is,  in  our  opinion,  far 
superior  to  any  thing  that  has  fallen  under  our  notice. 

CHARLES  MORILL,  8th  Grammar  School. 

NASON  H.  MORSE,  4th        do.  do. 

O.  H.  MORILL,  6th        do.  do. 

JONA.  KIMBALL,      3d         do.  do. 

PERLEY  BALCH,      1st         do.  do, 


From  the  School  Committee  of  Lowell. 

At  a  meeting  of  the  School  Committee  of  Lowell,  held  March  15th, 
1847,  it  was 

Voted,  That  F.  A.  Adams's  Arithmetic  be  adopted  for  the  High  School. 
FREDERICK  PARKER,  Secretary. 

At  a  meeting  of  the  School  Committee  of  Lowell,  held  May  2d,  1848, 
the  principals  of  several  of  the  Grammar  Schools  having  expressed  a  de- 
sire to  use  F.  A.  Adams's  Arithmetic  in  their  schools,  on  motion,  it  was 

Voted,  That  F.  A.  Adams's  Arithmetic  be  adopted,  to  be  used  as  a 
text-book  in  any  Grammar  School  the  principal  of  which  may  be  desirous 
of  using  it. 

FREDERICK  PARKER,  Secretary. 


RECOMMENDATIONS.  1 1 

From  Mr.  A.  Parish,  Principal  of  Springfield  High  School. 

Mr.  F.  A.  Adams. 

Dear  Sir: — I  have  just  completed  the  perusal  of  your  Arithmetic,  de- 
signed to  impart  to  pupils  the  power  of  solving-,  mentally,  problems  in- 
volving large  numbers.  The  misgivings  which  I  felt  at  the  commence- 
ment of  my  investigation  vanished  as  I  advanced.  While  the  leading 
feature  of  the  work  resembles,  in  some  respects,  Colburn's  Unrivalled  pro- 
duction, in  its  processes  and  application  of  principles,  it  seems  to  open  to 
the  pupil  an  entirely  new  field  for  mental  calculations.  The  great  facility 
with  which  large  numbers  are  rapidly  disposed  of  without  encumbering 
the  mind,  is  an  element  in  the  work  which  must  contribute  greatly  to  the 
satisfaction  as  well  as  success  of  the  scholar. 

The  very  clear  explanation  and  illustration  of  principles ;  the  original 
and' varied  character  of  the  problems,  happily  graduated  from  the  simple 
to  the  more  difficult ;  the  adaptation  of  the  whole  matter,  both  to  produce 
mental  discipline  and  due  preparation  for  business,  must  commend  the 
book  forcibly  to  the  attention  of  teachers  who  desire  to  employ  the  most 
effectual  aids  in  their  profession. 

A.  PARISH. 


V 


YB 


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THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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V 


THOA[Aa^OWPERTHWJffr&  CO.,  ! 

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